Rookie Galchenyuk -- the third overall pick of the 2012 draft -- got his second career goal less than 2 minutes into the final frame to break a 1-1 tie.
Max Pacioretty also scored and Raphael Diaz added an empty-netter as the Northeast Division-leading Canadiens racked up their fifth straight win.
Carey Price earned the victory with 24 saves.
Anton Stralman scored for the Rangers, who were playing without star Rick Nash. The high-scoring forward was scratched before the game with an unspecified injury.
Henrik Lundqvist turned aside 15-of-17 shots as New York suffered its first regulation loss in six games.