I was reading Titus and noticed that Titus 2:11 ("For the grace of God has appeared, bringing salvation for all people") bears some resemblance to the first two lines of the hymn: "Salvation unto us has come / By God's free grace and favor."
About a week later, KFUO tweeted the ninth verse, and I realized that the lines "And by its fruits true faith is known, / With love and hope increasing" echo the qualities listed in 1 Corinthians 13:13: "So now faith, hope, and love abide, these three; but the greatest of these is love." I also noticed that the lines "Works serve our neighbor and supply / The proof that faith is living" seem to refer to the last part of James 2. Verse 17 is the most similar, although it has a slightly different point of view: "So also faith by itself, if it does not have works, is dead."