Your statement doesn't make sense. Allowing Jesus to save you does not make you your own savior, it makes Jesus your Savior. Accepting a free gift does not make you the giver of the gift. If you are offered a gift you didn't earn and don't deserve and you accept it, that doesn't mean you earned it. Accepting Christs gift isn't the payment for our sins, the payment was death on the cross.
Allowing Jesus to save you does not make you your own savior
Sure it does because it gives you the final say-so in determining whether or not you are saved. God becomes passive, waiting for men to make up their minds.
That's not the God of Scripture.
"Blessed is the man whom thou choosest, and causest to approach unto thee" -- Psalm 65:4
If you are offered a gift you didn't earn and don't deserve and you accept it, that doesn't mean you earned it.
It means you are smart enough and wise enough and pious enough to know the worth of the gift and the depth of your own insufficiency. But the natural man does not possess the ability to know those things. Only the spiritual man can know the things of God, can feel true sorrow of his sins and repent of them, and this ability occurs after that man has been reborn by the Holy Spirit, according to the will and purpose of God, and not according to men's own running or willing.
"Therefore said I unto you, that no man can come unto me, except it were given unto him of my Father." -- John 6:65
If you accept the fact that you were born with a sinful nature and that your nature was in rebellion against God, then you would naturally be incapable of accepting God's gift because the thought of that alone would be as pleasing to your natural heart as the thought of eating a bowl of strawberries covered in syrup of ipecac.
In our natural state we are all Christopher Hitchens. We are born with natural enmity against God. So therefore in order for us to accept this Gift, our nature must be changed from one of enmity against God to Love and Fear of God. If God makes the change, then God and God alone becomes our Savior. However if God does not change our nature and somehow we can muster up within our own selves and through our own nature the ability to turn our enmity against God into a profound love of God such that we are now willing to turn from our sins voluntarily and repent and follow Christ, then we are in effect our own saviors.
In other words, what you are suggesting is that Christ is the General Savior (i.e., He made it "possible" for you to be saved), but you are your own "Particular Savior", i.e., you changed your own nature and through your own sacrifice or work, i.e., turning to Christ (violating your own nature) and repenting (again violating your own nature) and developing within your own heart (that black heart of stone you were born with) a profound Love of God and acknowledgment of his tremendous sacrifice on your behalf (something that is basically anathema to our sin nature) and then to seal the deal, you accepted Christ and secured your rightful place in Heaven.
You are to be commended then for your personal insight, your personal work in changing your stone heart into one of flesh, your personal acknowledgment of your sins and your personal acceptance of his gift. Congratulations on being your own Particular Redeemer.
On the other hand, if all those changes were the result of God changing your nature and God turning your heart of stone into a heart of flesh and God turning your natural enmity against him to a profound Love and a deep sorrow for your sins, then God alone is your Savior as Christ not only died for your sins, but he changed your nature such that you not only had the ability to turn from your sins to follow him, but that it was inevitable that you would.
So if you want to argue that your salvation was something you "achieved" by you somehow changing or rebelling against your own sinful nature without God intervening and making it happen, then you saved yourself.
But I think deep down you must recognize that you didn't save yourself (unless you are a full blown Pelagian). It was either your work or the work of God. I suspect that if you are truly saved, that you will recognize that it was entirely the work of God and that it was a miracle that you ever turned to Him and that you could possibly Love him. But if it was just in your nature to turn to him and love him and God did not violate your free will to bring you to him, then I don't think you can escape the conclusion that you saved yourself.
So which was it?