Excellent question. Good works can never pay the penalty for sin and sin is what separates us from God.
The wages of sin is DEATH. But the gift of God is eternal life through Jesus Christ our Lord. (Rom. 6:23)
As explained in post above -- our salvation is from Christ's sacrifice only. It is God's grace that saves us, we cannot save ourselves, that is Church teaching since Apostolic times.
Our works are just following God's words to "If any man would come after me, let him deny himself and take up his cross and follow me" (Mt 16:24) and our works are our following scripture which tells us that we must work out our own salvation (which does not mean we can save ourselves, yet points out that we can lose the freely granted grace of Salvation that Jesus Christ gave us by His sacrifice on the Cross)