The Canucks won for the fourth time in five games and ended the Blues' four-game winning streak.
Burrows opened the scoring 2:48 into the game and his tip-in with 2:08 left in the second forged a 2-2 tie. He has 18 goals for the season.
Arnott tallied 62 seconds after Burrows' first goal and then gave St. Louis a 2-1 lead 6:27 into the middle period.
Roberto Luongo made 31 saves for the Canucks. Elliott allowed the three goals on 21 shots.