Alex Galchenyuk, Erik Cole and Lars Eller each scored a goal and added an assist while Carey Price made 17 saves for his second shutout of the season.
All three goals came in the second period.
Cole was first to get on the score sheet with a clever wrist flick early in the period before Galchenyuk doubled the advantage with a slick move near the net.
Eller closed the scoring with a powerplay slapshot with less than 30 seconds remaining in the period.
The loss was not the only concern for the Rangers who lost key defensemen Ryan McDonagh and Dan Girardi to injury.
The Canadiens improved to 12-4-2 with the win to be two points clear of the Pittsburgh Penguins, Ottawa and the New Jersey Devils in the East.
The Rangers dropped to 8-7-2 and are struggling in fourth spot in the Atlantic Division.
(Reporting by Ben Everill in Tucson; Editing by Greg Stutchbury)