Short answer
Yes. The letter to the Hebrews says Jesus prayed to the One able to save him from death, and that he was heard. At the grave of Lazarus, Jesus told God (﷾) that He hears him always. In the garden of Gethsemane he prayed three times that the cup of suffering would pass from him. Luke also says that an angel came from heaven to strengthen him. The Psalms promise that when the righteous cry, the Lord hears and delivers them. So a prayer to be saved from death that is heard is a prayer answered by rescue. The Qur'an says what happened: they did not kill him or crucify him, and Allah (﷾) raised him up to Himself.
Full explanation
What he asked for in Gethsemane
On the night of his arrest, Jesus asked that the cup pass from him. He reminded God (﷾) that all things are possible to Him. He prayed three times. Luke says an angel came from heaven to strengthen him. These two verses of Luke, 22:43-44, are missing from several of the earliest manuscripts, the oldest hand-written copies of Luke. But they stand in the King James Version, and most modern Bibles still print them, some in brackets.
“And he went a little further, and fell on his face, and prayed, saying, O my Father, if it be possible, let this cup pass from me: nevertheless not as I will, but as thou wilt.”
“And he said, Abba, Father, all things are possible unto thee; take away this cup from me: nevertheless not what I will, but what thou wilt.”
“Saying, Father, if thou be willing, remove this cup from me: nevertheless not my will, but thine, be done.”
“And he left them, and went away again, and prayed the third time, saying the same words.”
“And there appeared an angel unto him from heaven, strengthening him. And being in an agony he prayed more earnestly: and his sweat was as it were great drops of blood falling down to the ground.”
Heard, and was able to save him
Hebrews, a letter in the New Testament, names God as the one "able to save him from death". It says the prayer was heard. The Psalms, the Jewish book of songs and prayers, promise that when the righteous cry, the Lord hears and delivers them. In John, the man Jesus healed of blindness states the same principle: God (﷾) hears the one who worships Him and does His will. Jesus taught that no father answers a son's request for bread with a stone.
“Who in the days of his flesh, when he had offered up prayers and supplications with strong crying and tears unto him that was able to save him from death, and was heard in that he feared;”
“The righteous cry, and the Lord heareth, and delivereth them out of all their troubles.”
“Now we know that God heareth not sinners: but if any man be a worshipper of God, and doeth his will, him he heareth.”
“Or what man is there of you, whom if his son ask bread, will he give him a stone? Or if he ask a fish, will he give him a serpent? If ye then, being evil, know how to give good gifts unto your children, how much more shall your Father which is in heaven give good things to them that ask him?”
At the grave of Lazarus, Jesus says to God (﷾): “I knew that thou hearest me always”.
What did he ask for?
In Gethsemane he prayed, “O my Father, if it be possible, let this cup pass from me: nevertheless not as I will, but as thou wilt.” He prayed three times.
What came?
Luke, as most Bibles print it, adds: “there appeared an angel unto him from heaven, strengthening him”.
Was the prayer answered?
Hebrews says he prayed “unto him that was able to save him from death, and was heard”.
A prayer to be saved from death that is heard is a prayer answered by rescue. The Qur'an says Allah (﷾) raised him up to Himself.
God (﷾) always heard him
At the tomb of Lazarus, Jesus thanks God (﷾) for hearing him and says God (﷾) hears him always. By his own words, his prayers were not turned away.
“Then they took away the stone from the place where the dead was laid. And Jesus lifted up his eyes, and said, Father, I thank thee that thou hast heard me. And I knew that thou hearest me always: but because of the people which stand by I said it, that they may believe that thou hast sent me.”
How Christians read the prayer
Christians read Hebrews 5:7 as a prayer answered after death. They say God (﷾) brought Jesus out of death by raising him on the third day. They also point out that Jesus ended each prayer in Gethsemane by yielding to the Father's will. And they say that in John, Jesus says he came for this very hour.
“Who in the days of his flesh, when he had offered up prayers and supplications with strong crying and tears unto him that was able to save him from death, and was heard in that he feared;”
“He went away again the second time, and prayed, saying, O my Father, if this cup may not pass away from me, except I drink it, thy will be done.”
“Now is my soul troubled; and what shall I say? Father, save me from this hour: but for this cause came I unto this hour.”
What the Qur'an says happened
The Qur'an says Jesus was neither killed nor crucified, and that Allah (﷾) raised him up to Himself, as He had promised him. It tells the same kind of story about Jonah. He called out in the dark, God (﷾) listened, and God (﷾) delivered him.
“That they said (in boast), "We killed Christ Jesus the son of Mary, the Messenger of Allah";- but they killed him not, nor crucified him, but so it was made to appear to them, and those who differ therein are full of doubts, with no (certain) knowledge, but only conjecture to follow, for of a surety they killed him not:-”
“Nay, Allah raised him up unto Himself; and Allah is Exalted in Power, Wise;-”
“Behold! Allah said: "O Jesus! I will take thee and raise thee to Myself and clear thee (of the falsehoods) of those who blaspheme; I will make those who follow thee superior to those who reject faith, to the Day of Resurrection: Then shall ye all return unto me, and I will judge between you of the matters wherein ye dispute.”
“And remember Zun-nun, when he departed in wrath: He imagined that We had no power over him! But he cried through the depths of darkness, "There is no god but thou: glory to thee: I was indeed wrong!" So We listened to him: and delivered him from distress: and thus do We deliver those who have faith.”
Verse by verse
Hebrews 5:7 — Epistle to the Hebrews, chapter 5, verse 7
“Who in the days of his flesh, when he had offered up prayers and supplications with strong crying and tears unto him that was able to save him from death, and was heard in that he feared;”
Cited for
"Save him from death" means "out of death". God heard the prayer by raising Jesus after he died.
The response
The prayer in Hebrews, with strong crying and tears, matches Gethsemane, where he asked that the cup pass from him. He did not ask to die and be raised. The writer of Hebrews also believed that Jesus died and was brought again from the dead, as the churches of his day taught. But the report he keeps of the prayer is plain. Jesus asked the One able to save him from death, and he was heard. On its own words, that is a prayer answered by rescue, which is what the Qur'an says happened.
Where it lands — Hebrews says the prayer to be saved from death was heard, and a prayer to be spared death is answered only if death is spared.
Matthew 26:42 — Gospel of Matthew, chapter 26, verse 42
“He went away again the second time, and prayed, saying, O my Father, if this cup may not pass away from me, except I drink it, thy will be done.”
Cited for
Jesus submitted to the Father's will, and the Father's will was the cross. The request was conditional.
The response
Submitting to God's (﷾) will is how every prophet prays. It does not make a heard prayer into a refused one. Hebrews reports the outcome in one word: he was heard. Luke also says that God (﷾) sent an angel to strengthen him. And Jesus taught that the Father gives good things to those who ask.
Where it lands — The prayer yielded to God's (﷾) will, and the New Testament says God (﷾) heard it.
John 12:27 — Gospel of John, chapter 12, verse 27
“Now is my soul troubled; and what shall I say? Father, save me from this hour: but for this cause came I unto this hour.”
Cited for
In John, Jesus refuses to pray "save me from this hour", because he came for it.
The response
Matthew, Mark and Luke all record that he did pray for the cup to pass, and Matthew says he prayed three times. Hebrews also says he prayed to be saved from death. Hebrews was written to explain his priesthood. John's verse cannot erase what three Gospels and Hebrews report.
Where it lands — The New Testament records that he prayed to be saved from death, and that he was heard.
Perspectives
Mainstream Christian
God answered the prayer by raising Jesus from the dead. Jesus asked for the cup to pass only if the Father willed it, and he went to the cross willingly.
Hebrews 5:7 means that God saved Jesus out of death by raising him, not that God kept him from dying.
Rebuttal
The prayer in Hebrews, with strong crying and tears, matches Gethsemane, where he asked that the cup pass from him. He asked to be spared, not to die and be raised. Hebrews says he was heard, and a prayer to be spared death is answered by sparing death.
In Gethsemane Jesus placed the Father's will above his own request.
Rebuttal
Putting God's (﷾) will first is how every prophet prays. It does not turn a heard prayer into a refused one. Hebrews says he was heard, and in the King James text of Luke, God (﷾) sent an angel to strengthen him.
In John, Jesus says he came for this very hour.
Rebuttal
John's verse cannot erase what three Gospels and Hebrews report. Matthew, Mark and Luke all say he prayed for the cup to pass. Hebrews says he prayed to be saved from death.
The Islamic perspective
God (﷾) heard the prayer of His prophet and saved him from death. The New Testament's own record of that prayer, and of its being heard, points to the Qur'an's answer.
Jesus said that God (﷾) hears him always.
In Gethsemane Jesus asked God (﷾) to let the cup pass from him, and he returned to prayer three times.
Most Bibles, the King James Version among them, print a verse in Luke about this. It records that an angel from heaven came to strengthen him as he prayed.
Hebrews says he prayed to the one able to save him from death, and was heard.
Taken together, these verses point to a prayer to be spared death that God (﷾) heard and answered.
The Qur'an teaches that Jesus was not killed or crucified, and that Allah (﷾) raised him to Himself.