The Canadiens won for the first time in three games and extended the Sabres' losing streak to four.
Montreal's Carey Price gave up the game-tying goal to Tyler Ennis with 8:05 to play in the third period, but he shut down Brad Boyes and Jason Pominville in the shootout to secure the victory. Price finished with 27 saves.
Both sides scored twice in the opening period and Montreal took a 3-2 lead in the second on Chris Campoli's goal with 8:32 left in the stanza.
Ryan Miller stopped 16-of-19 shots in regulation and overtime for the Sabres.