"Carey Price kept us in, in the first period when we needed a goaltender to keep us in and at least try and win ourselves a hockey game," Canadiens coach Claude Julien said.
Galchenyuk tied the game 1-1 when he scored on a rush at 5:19 of the second period, beating Ward over the glove for his ninth of the season.
Ward, who began the season as the backup after the Hurricanes acquired Scott Darling in a trade with the Chicago Blackhawks on April 29 and signed him to a four-year, $16.6 million contract, has started five of the past seven games.
"We knew going into it, it was going to be fairly low scoring facing Carey Price, who in my opinion is the best goalie in the world," Ward said. "We got two goals today, and fortunately that was good enough to win."
Julien said, "When you look at tonight, [forward Jonathan] Drouin had a bunch of turnovers, and you can't play that way. You need to make sure that you're managing the puck well, and maybe instead of looking for those fancy passes, sometimes you've got to do it the right way and throw some pucks at the net."