Absolutely! Which is why Jesus is the Savior. Scripture would not refer to HIM as the Savior if He didn't save anyone, Scripture would call US the Savior (of self anyway) if self saves self by what self does.
While the Bible itself speaks of being saved "by grace" and "by Jesus" and "by faith" actually all 3 of these are aspects of justification. Theology expresses this as one, inseparable truth: SOLA GRATIA - SOLUS CHRISTUS - SOLA FIDE (by the unmerited favor/gifts of God - through the life/death/resurrection of Christ as THE one and only, all sufficient Savior - which is apprehended by faith). John 3:16, "For God loved the world so that He gave His only-begotten Son that whosoever believes in Him will not perish but have eternal life." All this is God's work, for God is our Savior, not self. All other religions argue that "God helps those who help themselves" so that no Savior is needed - just lots of divine help, lots of time, and perhaps a good example/model/teacher. Christianity is unique in that Christianity argues we need to be RESCUED, SAVED and we can't do it, God must. And did.
Now, once justified (in this narrow or initial sense), once we are His own, we are Called by God to great things (as most parents call their children). To be holy as God is. To be as perfect as God is. To be as loving as God is. And much more. But these are not what "saves" or justifies or enlivens us, this is what is called for BECAUSE we are spiritually alive, BECAUSE we are His own. I'm alive.... and so I should be a moral, responsible, contributing person but being so is not why I have life, it's my having life that means I should be those things.
To the question: YES, we are saved by works. The question is this: WHOSE? If it's MINE then I'm the Savior. If it's JESUS' then He is the Savior. And if it's a mixture, then self is the PART Savior and Jesus is the PART Savior (but likely, Jesus has the part that actually doesn't save anyone).
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