Christian Thomas set up the winner after scoring the tying goal late in the second period. Jiri Sekac had Montreal's other goal.
Ryan Spooner and Matt Fraser scored for Boston.
Only 1:17 in, first round pick Nikita Scherbak's blind back pass was picked off by Spooner, who went in alone to beat Dustin Tokarski.
The teams exchanged power-play goals. Fraser scored at 9:12, and Sekac beat Niklas Svedberg with a shot off of David Krejci's stick at 19:52.
Thomas tied it at 16:49.
Michael McCarron, Montreal's 2013 first-round pick, sustained an arm injury late in the second period when he went to the net.