You really think there's much difference? The New Testament says nobody was saved under the law. So keeping the law didn't save him. What did? He admitted his sin to the Lord and asked for mercy. The main difference between the Old and the New covenant is that prior to Jesus, They tried to keep the law and they trusted in God for forgiveness and salvation when they couldn't, and afterwards they understood what God did to provide that salvation and how much he love us.
It still comes down to faith in God and being forgiven.
Did animal sacrifices save the Hebrews? Passage Psalm 51:16-17: 16For thou desirest not sacrifice; else would I give it: thou delightest not in burnt offering. 17The sacrifices of God are a broken spirit: a broken and a contrite heart, O God, thou wilt not despise.
I guess you missed my point. Until Christ’s sacrifice of His life and His resurrection from the dead, the New covenant was not in place. Christ had the power and authority to save anyone He wished while on the Earth. Under the New covenant, one is saved by accepting the gift of salvation by being obedient to the Gospel.