There's more diversity on this point within Christian history, tradition, and doctrine than I think we realize sometimes.
Isaiah 53:5
"But he was wounded for our transgressions, he was bruised for our iniquities: the chastisement of our peace was upon him; and with his stripes we are healed."
Jesus suffering for our sin doesn't have to mean that God's wrath was satisfied by punishing Him and making Him suffer instead of us. The Gospel can and has also been articulated something like this:
By sending Christ to earth as a man, the Father condemned Christ to death, the condemnation of man due to his sinful rebellion against God. Perfect God stepped into fallen world and subjected Himself to its wickedness and suffered the wages of sin (death), which were not His to suffer.
However, it was not His suffering which appeased God, but rather His righteousness and perfect obedience. He lived the perfect life that I could not live. He was wholly faithful and wholly obedient, and His death on the Cross perfected His obedience as a man, as there is no greater or fuller obedience that one can offer God besides the submission of one's entire life to Him--unto death. Jesus had to die in order to redeem man because in order to redeem man He had to live fallen man's life to the fullest and to the end, which includes death. Instead of avoiding it, instead of seeking His own will, He lived it, suffered it, and triumphed over it, the grave not being able to hold Him because of His perfect righteousness.
As a believer in Christ, I am not found righteous because Jesus was punished instead of me, but rather because just as I fell in the first Adam, in Christ the second Adam, I will be made whole. We can become part of the second Adam (Christ) by uniting ourselves to Him in faith. We as believers become His by acknowledging that our lives do not please God and that only Christ was able to please Him fully. Therefore, we look to Him, believe on Him, hide ourselves in Him, and await His coming again, when we will be raised to reign with Him, as His Bride, His Body, having been enveloped into Himself.