Carey Price made 32 saves during regulation and then stopped six of eight Pittsburgh shooters in the tiebreaker, including Jason Williams on the Penguins' final attempt.
Evgeni Malkin used a spin-around move to convert Pittsburgh's second attempt after Rene Bourque shot and scored on Montreal's first.
Montreal's Lars Eller and Pittsburgh's Chris Kunitz both scored in the fifth round.
The Canadiens, who beat Winnipeg 3-0 on Sunday, won for just the second time in nine shootouts this season.
Louis Leblanc scored in the second period to put Montreal up 1-0. Eller restored the Canadiens' lead at 2-1 with his 11th goal 3 minutes into the third.
James Neal scored his 28th goal 7:32 into the third as Pittsburgh erased a pair of one-goal deficits in the period. Evgeni Malkin assisted on the goal to extend his NHL lead to 62 points.
Pascal Dupuis drew Pittsburgh even at 1 early in the third when he drove the net to score a short-handed goal.
Marc-Andre Fleury stopped 28 shots for the Penguins, who fell to 7-3 in shootouts.
Pittsburgh coach Dan Bylsma called his timeout with a faceoff in Montreal's zone and 18.3 seconds left in overtime.
Penguins defenseman Kris Letang jumped on Montreal's P.K. Subban as time expired after Malkin weathered three straight hits from the Canadiens.
Price made a pad save on Kunitz's quick shot 7 minutes into the second on a setup off a faceoff won by Malkin in the Canadiens' zone.
The Montreal goalie lost his bid for a second straight shutout 1:03 into the third when Dupuis was credited with a short-handed goal that tied it at 1. Joe Vitale's centering pass from the right side went in off Dupuis' right skate as he drove to the net. A brief video review upheld the call.
Fleury came up with a few big stops of his own, including Leblanc on a partial breakaway 13 minutes in. He also turned aside Plekanec's shot on a short-handed 3-on-1 early in the second.
Plekanec was denied by Fleury again when he came out of the penalty box to take a breakaway pass right after Subban stopped Malkin's shot at an otherwise open net. Price was caught out of position when he fell after coming out to play the puck late in the Penguins' power play.
Luck was against Fleury as the Canadiens regained the lead shortly after Kunitz's tying goal.
Eller took a shot from the left side just over the blue line that struck the right post. The puck bounded back off Fleury's backside and across the goal line to give Montreal its second lead.
Leblanc scored his third goal 11:21 into the second. It was his first in four games since he was recalled from AHL Hamilton.
NOTES: Williams dressed after he was recalled from AHL Scranton/Wilkes-Barre earlier in the day. He had one goal and one assist in seven games with Pittsburgh earlier this season. ... Canadiens LW Travis Moen returned after missing five games because of an upper-body injury. ... Montreal D Yannick Weber went to the dressing room after Kunitz's check sent him into the boards 3:48 into overtime.
- Pittsburgh Penguins
- Evgeni Malkin
- Chris Kunitz
- the Penguins