Max Pacioretty and Alex Galchenyuk added a goal and assist each for the Canadiens, who received additional tallies from Tomas Plekanec and David Desharnais.
Andrei Markov added three helpers, Lars Eller and Dale Weise assisted on a pair of scores apiece, while Dustin Tokarski stopped 23 shots in the victory.
Derek Stepan returned from a broken jaw to net a pair of goals for the Rangers, who battled back from a 4-1 deficit but get another chance to close out the series on home ice in Game 6 on Thursday.
Chris Kreider contributed a goal and three assists and Rick Nash also tallied. Cam Talbot surrendered two goals on eight shots in relief of Henrik Lundqvist, who exited after yielding four goals on 19 chances.