Short answer
Islam is not new. The Qur'an teaches that every prophet, from Adam to Muhammad (ﷺ), brought the same faith: worship the one God (﷾) alone and follow His messenger. Islam means submission to God (﷾), so every believer who did this was a Muslim in that sense. The Qur'an calls Abraham one who submitted, and Jesus' disciples asked to be witnessed as Muslims. Each prophet came to his own people with the scripture of his time, such as the Torah of Moses and the Injil (Gospel) of Jesus. A believer in Moses' day followed Moses. A believer in Jesus' day followed Jesus. Muhammad (ﷺ) is the last prophet, sent to all mankind. So today a person submits to God (﷾) by following him and the Qur'an.
Full explanation
One message from the first prophet to the last
The Qur'an says that Allah (﷾) sent no messenger without revealing to him that there is no god but He, so worship Him alone. It says a messenger was sent to every people with the same command: serve Allah (﷾) and avoid false gods. It names the line of that one religion. What was enjoined on Noah, Abraham, Moses and Jesus is what was revealed to Muhammad (ﷺ). Noah's first words to his people were: worship Allah (﷾), you have no other god but Him. The line begins with Adam, whom Allah (﷾) chose. The Qur'an also says Allah (﷾) drew out the descendants of the children of Adam. He asked them, "Am I not your Lord?", and they answered yes. The Prophet (ﷺ) put it simply: the prophets are brothers with one father and different mothers. Their religion is one.
“Not a messenger did We send before thee without this inspiration sent by Us to him: that there is no god but I; therefore worship and serve Me.”
“For We assuredly sent amongst every People a messenger, (with the Command), "Serve Allah, and eschew Evil": of the People were some whom Allah guided, and some on whom error became inevitably (established). So travel through the earth, and see what was the end of those who denied (the Truth).”
“The same religion has He established for you as that which He enjoined on Noah - the which We have sent by inspiration to thee - and that which We enjoined on Abraham, Moses, and Jesus: Namely, that ye should remain steadfast in religion, and make no divisions therein: to those who worship other things than Allah, hard is the (way) to which thou callest them. Allah chooses to Himself those whom He pleases, and guides to Himself those who turn (to Him).”
“We sent Noah to his people. He said: "O my people! worship Allah! ye have no other god but Him. I fear for you the punishment of a dreadful day!”
“Allah did choose Adam and Noah, the family of Abraham, and the family of 'Imran above all people,-”
“When thy Lord drew forth from the Children of Adam - from their loins - their descendants, and made them testify concerning themselves, (saying): "Am I not your Lord (who cherishes and sustains you)?"- They said: "Yea! We do testify!" (This), lest ye should say on the Day of Judgment: "Of this we were never mindful":”
Abu Hurayra reports that the Prophet (ﷺ) said he is the closest of all people to Jesus son of Mary, in this world and the next; the prophets are brothers with one father and different mothers, and their religion is one.
Muslims before Muhammad (ﷺ)
Muslim is an Arabic word for one who submits to God (﷾). In that sense the Qur'an uses it for believers long before the Prophet Muhammad (ﷺ). It says Abraham was not a Jew nor a Christian, but upright, one who submitted to Allah (﷾). Abraham and Jacob told their sons to die only in submission to Him. Moses told his people to trust in Allah (﷾) if they had submitted to Him. Jesus' disciples said, "We have faith, and do thou bear witness that we bow to Allah as Muslims." The Qur'an says Allah (﷾) named the believers Muslims "both before and in this" revelation.
“Abraham was not a Jew nor yet a Christian; but he was true in Faith, and bowed his will to Allah's (Which is Islam), and he joined not gods with Allah.”
“Behold! his Lord said to him: "Bow (thy will to Me):" He said: "I bow (my will) to the Lord and Cherisher of the Universe." And this was the legacy that Abraham left to his sons, and so did Jacob; "Oh my sons! Allah hath chosen the Faith for you; then die not except in the Faith of Islam." Were ye witnesses when death appeared before Jacob? Behold, he said to his sons: "What will ye worship after me?" They said: "We shall worship Thy god and the god of thy fathers, of Abraham, Isma'il and Isaac,- the one (True) Allah: To Him we bow (in Islam)."”
“Moses said: "O my people! If ye do (really) believe in Allah, then in Him put your trust if ye submit (your will to His)."”
“"And behold! I inspired the disciples to have faith in Me and Mine Messenger: they said, 'We have faith, and do thou bear witness that we bow to Allah as Muslims'".”
“And strive in His cause as ye ought to strive, (with sincerity and under discipline). He has chosen you, and has imposed no difficulties on you in religion; it is the cult of your father Abraham. It is He Who has named you Muslims, both before and in this (Revelation); that the Messenger may be a witness for you, and ye be witnesses for mankind! So establish regular Prayer, give regular Charity, and hold fast to Allah! He is your Protector - the Best to protect and the Best to help!”
Each prophet for his people and his time
Every messenger spoke in the language of his own people, to make the message clear to them. Allah (﷾) gave Moses the Torah, "guidance and light", and the prophets after him judged by it. He sent Jesus with the Injil, the Gospel, confirming the Torah before him. Jesus himself said he was sent only to the lost sheep of the house of Israel. So a believer in Moses' time followed Moses and the Torah, and a believer in Jesus' time followed Jesus. The laws could differ from one prophet to the next. The Qur'an says that to each it gave a law and a way, while the faith in the one God (﷾) stayed the same.
“We sent not a messenger except (to teach) in the language of his (own) people, in order to make (things) clear to them. Now Allah leaves straying those whom He pleases and guides whom He pleases: and He is Exalted in power, full of Wisdom.”
“It was We who revealed the law (to Moses): therein was guidance and light. By its standard have been judged the Jews, by the prophets who bowed (as in Islam) to Allah's will, by the rabbis and the doctors of law: for to them was entrusted the protection of Allah's book, and they were witnesses thereto: therefore fear not men, but fear me, and sell not my signs for a miserable price. If any do fail to judge by (the light of) what Allah hath revealed, they are (no better than) Unbelievers.”
“And in their footsteps We sent Jesus the son of Mary, confirming the Law that had come before him: We sent him the Gospel: therein was guidance and light, and confirmation of the Law that had come before him: a guidance and an admonition to those who fear Allah.”
“But he answered and said, I am not sent but unto the lost sheep of the house of Israel.”
“To thee We sent the Scripture in truth, confirming the scripture that came before it, and guarding it in safety: so judge between them by what Allah hath revealed, and follow not their vain desires, diverging from the Truth that hath come to thee. To each among you have we prescribed a law and an open way. If Allah had so willed, He would have made you a single people, but (His plan is) to test you in what He hath given you: so strive as in a race in all virtues. The goal of you all is to Allah; it is He that will show you the truth of the matters in which ye dispute;”
The last prophet, for everyone
The Qur'an calls Muhammad (ﷺ) the Seal of the Prophets, the last of them. Earlier prophets were sent to their own people, but he was told to say: "O men! I am sent unto you all." He was sent to all mankind. The Qur'an he brought confirms the scriptures before it and guards over them. Muslims believe in all the prophets and make no distinction between them in faith. Following the last one is how a person now believes in them all.
“Muhammad is not the father of any of your men, but (he is) the Messenger of Allah, and the Seal of the Prophets: and Allah has full knowledge of all things.”
“Say: "O men! I am sent unto you all, as the Messenger of Allah, to Whom belongeth the dominion of the heavens and the earth: there is no god but He: it is He That giveth both life and death. So believe in Allah and His Messenger, the Unlettered Prophet, who believeth in Allah and His words: follow him that (so) ye may be guided."”
“We have not sent thee but as a universal (Messenger) to men, giving them glad tidings, and warning them (against sin), but most men understand not.”
“To thee We sent the Scripture in truth, confirming the scripture that came before it, and guarding it in safety: so judge between them by what Allah hath revealed, and follow not their vain desires, diverging from the Truth that hath come to thee. To each among you have we prescribed a law and an open way. If Allah had so willed, He would have made you a single people, but (His plan is) to test you in what He hath given you: so strive as in a race in all virtues. The goal of you all is to Allah; it is He that will show you the truth of the matters in which ye dispute;”
“Say: "We believe in Allah, and in what has been revealed to us and what was revealed to Abraham, Isma'il, Isaac, Jacob, and the Tribes, and in (the Books) given to Moses, Jesus, and the prophets, from their Lord: We make no distinction between one and another among them, and to Allah do we bow our will (in Islam)."”
The Christian view
Christians teach that Jesus brought a new covenant. In their view the old covenant with Israel was fulfilled in him, and the Church is the new people of God (﷾), open to all nations.
“Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil. For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.”
What Jesus himself taught
Jesus said he had not come to destroy the Law or the prophets. Not one jot or tittle, he said, would pass from the Law. Asked for the first of all the commandments, he quoted Moses: "Hear, O Israel; The Lord our God is one Lord." That is the same first article of faith Moses taught, and the same one the Qur'an teaches. On the Qur'an's account, Jesus did not found a new religion. He renewed the one religion of the prophets before him.
“Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil. For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.”
“And Jesus answered him, The first of all the commandments is, Hear, O Israel; The Lord our God is one Lord:”
“Hear, O Israel: The Lord our God is one Lord:”
“The same religion has He established for you as that which He enjoined on Noah - the which We have sent by inspiration to thee - and that which We enjoined on Abraham, Moses, and Jesus: Namely, that ye should remain steadfast in religion, and make no divisions therein: to those who worship other things than Allah, hard is the (way) to which thou callest them. Allah chooses to Himself those whom He pleases, and guides to Himself those who turn (to Him).”
Perspectives
Christian
Christians teach that Jesus brought a new covenant that fulfilled the old. The Church, they say, is the new people of God.
Jesus came to fulfil the Law, so a new covenant began with him.
Rebuttal
In the same breath Jesus said not one jot or tittle would pass from the Law. His first commandment was Moses' own: "The Lord our God is one Lord". He kept the faith of the prophets before him, as the Qur'an says.
The Islamic perspective
Islam, submission to the one God (﷾), is the religion of every prophet from Adam to Muhammad (ﷺ). Each came to his people with the scripture of his time. Muhammad (ﷺ) is the last, sent to all mankind.
The Qur'an says every messenger was sent with the message that there is no god but Allah (﷾).
The Qur'an names the same religion for Noah, Abraham, Moses, Jesus and Muhammad (ﷺ).
The Qur'an calls Abraham one who submitted to Allah (﷾), and has Jesus' disciples ask to be witnessed as Muslims.
The Prophet (ﷺ) said the prophets are brothers whose religion is one.
Muhammad (ﷺ) is the Seal of the Prophets, sent to all mankind.