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.