Crucified for false accusation, how did Jesus' death turned into a redemptive sacrifice for our sins? In the OT God provided a way for His people to redeem their sins through burnt offering and sacrifices. In a way to demonstrate His grace and love for His people, and to give a glimpse of His salvation plan.

The circumstances in which Jesus was crucified is not as an offering to redeem anyone's sin. Instead it was intended as a punishment although for false accusations.
How did Jesus' death on the cross turned out to become a sacrifice to redeem sins?

While on the cross Christ pleaded the Father to forgive us. Then the whole of mankind's sins was on Him and the Father forsook Him. Am I right to say that, at that moment, through the Son's pleading, Son of God (God Himself) deomonstrates His great love. And in the act of shielding God's wrath Jesus took upon Himself the sins of all men. And the crucifixion becomes a redemptive sacrifice? 

So, though intended to be a punishment under man's law, Jesus, the trinity God made Himself a sacrificial lamb and die on the cross?
How did Jesus' death on the cross turned out to become a sacrifice to redeem sins?

While the Roman execution was taking place, there were other elements not as noticeable.


Jesus allowed himself to be put to death.



  John 10:17-18  Therefore doth my Father love me, because I lay down my
  life, that I might take it again.   No man taketh it from me, but I
  lay it down of myself. I have power to lay it down, and I have power
  to take it again. This commandment have I received of my Father.



The Father place the sins of the world on him.



  2 Corinthians 5:21  For he hath made him to be sin for us, who knew no
  sin; that we might be made the righteousness of God in him. 
  
  1 John 2:2  And he is the propitiation for our sins: and not for ours
  only, but also for the sins of the whole world.


If you look at the crucifixion and sees only a Roman execution, you miss the most important thing in the history of the world.