Boston won for the first time since a 2-1 shootout victory over San Jose on Jan. 14 and avoided tying the longest losing streak in franchise history set by the Bruins in their first season, 1924-25.
Rask, who recorded his fourth career shutout, made a number of big stops in the second period when Montreal outshot Boston 15-3 .
The Bruins scored a pair of goals 2:24 apart late in the first, including Adam McQuaid's first NHL goal and Sturm's first of two with 3.2 seconds remaining.
Sturm scored again midway through the third.
Jaroslav Halak stopped 24 shots in his fifth straight start for the Canadiens, who ended a three-game winning streak.
Bruins center David Krejci left in the first after playing three shifts for a total of 1:34. He did not return.
McQuaid, who made his NHL debut on Dec. 19 in a 2-0 loss in Toronto, scored 17:32 in. The 6-foot-4 defenseman's shot deflected into the net off Canadiens right wing Ryan White's stick.
Halak stopped Zdeno Chara's shot late in the first, but Sturm put away the rebound to put Boston up by two at 19:56.
Solid in the opening period, Rask shined in the second.
The Finnish goalie stuck out his right pad to stop Tomas Plekanec's shot on a breakaway 11:19 into the second, moments after he made a spectacular glove save on the Montreal center's slap shot from the slot.
Rask also robbed Plekanec with another pad save late in the middle period.
Sturm made it a three-goal lead 9:49 into the third when his long shot struck Canadiens defenseman Josh Gorges' stick on its way past Halak.
NOTES: The Bruins lost 11 straight from Dec. 3, 1924-Jan. 5, 1925. ... McQuaid was chosen 55th overall by Columbus in the second round of the 2005 draft. He had 10 points in 31 games with Providence of the AHL this season. ... Canadiens RW Matt D'Agostini was a healthy scratch.