What verses from the Bible claim that Jesus has justified and redeemed man by his sacrifice on a cross?
What you're talking about is called "substitutionary atonement" There are several references that support this viewpoint.
Surely he has borne our griefs and carried our sorrows; yet we esteemed him stricken, smitten by God, and afflicted. But he was pierced for our transgressions; he was crushed for our iniquities; upon him was the chastisement that brought us peace, and with his wounds we are healed. All we like sheep have gone astray; we have turned—every one—to his own way; and the Lord has laid on him the iniquity of us all.
2 Corinthians 5:21
For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God.
1 Peter 2:24
He himself bore our sins in his body on the tree, that we might die to sin and live to righteousness. By his wounds you have been healed.
1 Corinthians 15:3
For I delivered to you as of first importance what I also received: that Christ died for our sins in accordance with the Scriptures
Romans 4:23-25 (Jesus was delivered up for our sins, and raised for our justification)
But the words “it was counted to him” were not written for his sake alone, but for ours also. It will be counted to us who believe in him who raised from the dead Jesus our Lord, who was delivered up for our trespasses and raised for our justification.
1 John 4:10 (Propitiation: sufficient substitution for our sins)
In this is love, not that we have loved God but that he loved us and sent his Son to be the propitiation for our sins.
1 Peter 3:18 (Christ suffered for sins, though he was righteous, so that we may be considered righteous)
For Christ also suffered once for sins, the righteous for the unrighteous, that he might bring us to God, being put to death in the flesh but made alive in the spirit,
Galatians 3:13 (Christ became the curse for us, taking our penalty)
Christ redeemed us from the curse of the law by becoming a curse for us—for it is written, “Cursed is everyone who is hanged on a tree”
Galatians 2:20-21 (If salvation is by works, then Christ's death was for nothing)
I have been crucified with Christ. It is no longer I who live, but Christ who lives in me. And the life I now live in the flesh I live by faith in the Son of God, who loved me and gave himself for me. I do not nullify the grace of God, for if righteousness were through the law, then Christ died for no purpose.
Romans 3:21-26 (The righteousness is apart from the law, and given based on Christ's blood)
But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it— the righteousness of God through faith in Jesus Christ for all who believe. For there is no distinction: for all have sinned and fall short of the glory of God, and are justified by his grace as a gift, through the redemption that is in Christ Jesus, whom God put forward as a propitiation by his blood, to be received by faith. This was to show God’s righteousness, because in his divine forbearance he had passed over former sins. It was to show his righteousness at the present time, so that he might be just and the justifier of the one who has faith in Jesus.
There is John 3:16.
Then there was Jesus praying the day before the crucifixion that if there was another way that God would do that instead of the cross.
41 He withdrew about a stone’s throw beyond them, knelt down and prayed, 42 “Father, if you are willing, take this cup from me; yet not my will, but yours be done.” 43 An angel from heaven appeared to him and strengthened him. 44 And being in anguish, he prayed more earnestly, and his sweat was like drops of blood falling to the ground. LUKE 22:41-44
Jesus knew quite well what was coming and that it was necessary.
Paul says in Romans 5 that Christ's "righteous act" will save many. That act is the crucifixion.
18 Consequently, just as one trespass resulted in condemnation for all people, so also one righteous act resulted in justification and life for all people. 19 For just as through the disobedience of the one man the many were made sinners, so also through the obedience of the one man the many will be made righteous. Romans 5:18