Jesus can be our Savior to an even deeper degree, which means much more than just the forgiveness of sin.
The forgiveness of sin was also possible in the Old Testament. "They were cleansed from their sin on the day of Atonement." Leviticus 16:30. Of course, they needed continuous forgiveness for their sin, year after year. "Aaron came into the Holy Place with the blood of a young bull as a sin offering." Leviticus 16:3. In order to receive forgiveness, the children of Israel had to offer up the blood of bulls and lambs and goats as a sin offering.
More than forgiveness
Jesus came to offer us much more than forgiveness—He came to offer us total transformation! He understood that sin offerings could never lead people to a new and transformed life. He came to save us from our sin. This means that we do not have to go on committing sin year after year. The blood of bulls and goats could not take away sin, but Jesus could. (Hebrews 10:4)
Jesus came down to us with the same flesh as the children had. (Hebrews 2:14-16) He experienced the same temptations that we experience and overcame them while He was a man without sinning a single time! (Hebrews 4:15) How did He do this? Because of His obedience, God could condemn sin in His flesh while He was alive, before He died on Calvary. He died to His own will, long before they placed Him on the wooden cross. (Romans 8:3)
Our forerunner: a man like us
Jesus was made perfect through these everyday sufferings. (Hebrews 2:10) He did this so He could bring many of us to the same glory. He is our captain, and He has been completely saved from the sin in His flesh that He inherited through the fall. If we follow in His footsteps and let the Spirit do its work, sin can be put to death in our flesh, and we can cease from sin. (1 Peter 4:1)
We can stop sinning completely if we accept Jesus as our Savior. This means that it is possible to stop being angry, offended, and envious. We no longer have to be slaves to the "lust of our eyes." (1 John 2:16) Then we can be transformed into His image. (Romans 8:29) He became the firstborn of many brethren. What a Savior!