Galchenyuk's forehand beat Tuukka Rask over the right pad to give Montreal the 2-1 win on Monday night, ending the Bruins' streak at 12 games -- two wins shy of a franchise record.
The Canadiens led most of the game after Alexei Emelin's goal 6:39 into the first period but Patrice Bergeron tied it for Boston with 5:26 remaining in regulation on a deflected shot.
Peter Budaj made 28 saves in regulation and overtime and stopped Bergeron, Jarome Iginla, Brad Marchand and David Krejci in the shootout to help Montreal win for the fifth time in six games and move one point ahead of Tampa Bay for third place in the Eastern Conference.
The East-leading Bruins, who have already clinched a playoff spot, got their 104th point to break a tie with idle St. Louis for the NHL lead.
They hadn't lost since March 1 and were two wins shy of tying the longest run in team history set between Dec. 3, 1929 and Jan. 9, 1930.
Rask made 21 saves -- stopping all 17 shots he faced between the second period and overtime.