Rene Bourque and Max Pacioretty also scored for the Canadiens (16-9-3). Carey Price made 27 saves.
Steve Bernier and Michael Ryder scored for the Devils (11-12-5), and Martin Brodeur stopped 13 shots.
Ryder tied it at 2 with a slap shot that got past Price at 4:42 in the third. But Galchenyuk responded five minutes later, picking up his seventh goal after Alexei Emelin's shot landed on his stick.
It was New Jersey's first regulation loss at the Bell Centre since 2008.