When we truly turn to God in repentance through faith in Christ Jesus as Lord and Savior, at that moment in time, we are adopted into God's family as His child; we are saved.
What is often referred to as "the sinner's prayer" does not save us; it's just a way of sharing, with God, what's in our heart and on our mind.
The "sinner's prayer" is more for us, then anything else. It reminds us of that moment when we chose God, over self, through faith in Christ Jesus; it's a spiritual marker. Always remember that God searches beyond what we say; He searches our heart and mind to know the truth.
If in repentance you wish to ask God to forgive you of your sin, and through faith in Christ Jesus, you wish to ask God to save you, then please do so, God will answer your prayer.