The Chiefs' third-string quarterback, Quinn completed 15 of 21 passes, including a five-yard touchdown to Marvin Minnis in the third quarter that provided a 14-6 lead.
Starter Trent Green played briefly in the first quarter for the Chiefs and tossed a five-yard touchdown pass to Omar Easy.
Star running back Priest Holmes did not play for Kansas City (3-2), which hosts San Diego in its regular-season opener.
Jeff Wilkins kicked field goals of 34 and 39 yards in the first half for the Rams (1-3), who play at the New York Giants in their opener.
2014: The Year in Music [PHOTOS]