It's not exactly just three chords, with the chord on the first "peace" which according to one source is called a "G# Ø half dimº" chord, followed by the suspension on "heav-" aka "G add9 flat5 6th no3" [1]. There are some really cool passing chords in Silent Night.

[1] http://www.gootar.com/piano5/piano-chord.php