Salvation is about being forgiven and set free from sin. When someone is "saved," it means they've been delivered from the power of sin and most importantly from God's impending judgment. The truth is there is no way we could have been saved without God's righteousness, it was an absolutely needed. Without it, no one could be saved.
2 Corinthians 5:21 explains this to us:
"For he hath made him to be sin for us, who knew no sin; that we might be made the righteousness of God in him."
No person on Earth has the ability to save themselves. The demands of God's righteous law were completely fulfilled by Jesus Christ when He died on the cross. Because of His sacrifice, God's own righteousness was credited to our account the moment we were saved. Think of it like this: God's righteousness is the very principle by which He operates. Everything God does, He does according to righteousness.
The moment someone truly turns to the Lord, their sins are forgiven, and God's righteousness is instantly credited to their account. We are saved by God's grace, and through that grace, we are privileged to be made the righteousness of God. You can tell if a faith is genuine if it emphasizes this truth: that God's righteousness is put into us, because it's absolutely essential for our salvation.