Josh Anderson, Tyler Toffoli, Joel Armia, Paul Byron and Jeff Petry also scored for Montreal. Tomas Tatar had three assists and Carey Price stopped 28 shots.
The Canadiens were coming off a 4-3 overtime loss at Winnipeg on Thursday night and had had lost six of their last seven overall (1-2-4).
''I was really happy for them,'' Canadiens interim coach Dominique Ducharme said of his players. ''It won't be like that every night, but to be rewarded like our players go rewarded tonight, I think it's good for them because they put in the work.''
Mathieu Perrault spoiled Price's shutout bid with a power-play goal with 8:46 to go for Winnipeg, which had won six of its previous seven. Connor Hellebuyck gave up four goals on 19 shots before he was pulled in the second period. Laurent Brossoit stopped six of the nine shots he faced.
''We had a tough night here,'' Jets coach Paul Maurice said. ''We've got an entire hockey team that has a minus on them. Both goalies played, so at the very least it was well spread out.''
Leading 1-0 after 20 minutes, the Canadiens broke the game open in the second period with four goals in a 9:02 span.
After Jesperi Kotkaniemi won a faceoff in the Winnipeg end, Jets defenseman Tucker Poolman had a chance to clear but only sent the puck to the blue line. Shea Weber poked it back towards the slot where Toffoli's high wrist shot beat Hellebuyck for his 15th of the season at 7:03.
Failure to clear the puck cost Winnipeg again less than four minutes later with Gallagher knocking Phillip Danault's no-look cross-ice pass into a gaping net.
Gallagher, left alone on Hellebuyck's doorstep, then made it 4-0 with 5:35 left in the middle period with his 10th of the season after the Canadiens hemmed the Jets in their own zone.
''It was just one of those nights where the puck was bouncing all over for us and wasn't settling for us and it was settling for them,'' Hellebuyck said.
Brossoit took over in the Winnipeg goal and was promptly beaten by Armia's low wrist shot with 3:55 remaining after a Montreal rush up the ice that drew hardly a challenge.
Byron made it 6-0 at 4:20 of the third, backhanding a rebound home after Brossoit failed to handle a long-distance shot from Jake Evans. Seconds earlier Jets forward Trevor Lewis was hit on the hand by a Nate Thompson shot.
Petry added to the Jets' pain at 8:20 with a wrist shot through traffic from the blue line.
''They were a desperate team,'' Winnipeg's Blake Wheeler said. ''They're trying to get on a roll here and they made their opportunities count.''
The Canadiens got on the scoreboard in the first period as Anderson took advantage of a fortuitous bounce after Kotkaniemi, fighting for the puck in the Winnipeg end, fired the puck into the corner. Hellebuyck went behind the goal to corral the puck but it hit the board and bounced back in front, where Anderson knocked it into the open net with 4:31 let for his 10th.
Anderson returned to the Canadiens' lineup after a three-game absence due to a lower-body injury. He started on a line with Toffoli and Kotkaniemi.
SEEING FAMILIAR FACES
Saturday's game was the fifth under interim Canadiens coach Dominique Ducharme, who is now 2-1-2. Four of the games were against Winnipeg.
''You kind of get sick and tired of battling the same guys,'' Montreal defenseman Brett Kulak said before the game. ''We've played this team a lot right now. So yeah, there's a little bit of hatred for each other there, but I think it's good. And it brings the competitive level of the game up a little more.''
Jets: At Toronto on Tuesday night to open a three-game series against the Maple Leafs.
Canadiens: At Vancouver on Monday night to open a two-game series and six-game trip.
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