God didn't pass over their sins. Their sins were paid for by their Redeemer at the cross. Their justification was by faith (Read Hebrews 11).
The New Covenant comes because Jesus fulfilled the requirement of the Old Covenant. The New Covenant is affirmed at the last supper and implemented at the cross.
As to what Paul wrote in Romans, I wonder if you misunderstand Paul completely.
AustinC,
I'm not certain that Hebrews 11 is all that much helpful in supporting your thinking.
The last verse in that chapter states:
Now, the Scriptures do state that Christ came to fulfill the Law. What exactly does that mean? It means that the Law took no more dominion as the way to approach God (remember woman at the well asking about worship) but became a "school master" to direct people to Christ (Romans).
That was the "fulfillment" yet, we and everyone else is still accountable to the precepts for Christ stated that there are two principles to live by: Love God, Love your neighbor.
The New Covenant did not come because Christ fulfilled the requirements of the Old Covenant, but because the veil that was once a layer of separation between God's presence and humankind, is removed and we now have access to the throne of God by the veil of His body (Hebrews).
Hope this helps.