Corey Perry, Artturi Lehkonen and Nick Suzuki also scored for the Canadiens, who won their sixth straight game.
Adam Lowry scored for the Jets, who now face elimination on Monday.
Connor Hellebuyck turned aside 28 shots for Winnipeg.
Corey Perry opened the scoring 4:45 into the first period after pinning the Jets in their own end for over a half a minute before circling the net with the puck before firing the puck past Hellebuyck off of Jamie Benn's stick.
The goal was his 3rd of the playoffs.
Artturi Lehkonen scored his first of the playoffs 9:24 into the second period after Philip Danault hit the post.
Joel Armia scored shorthanded at 13:41 after Paul Stastny gave the puck away attempting to get the puck to the point.
Armia intercepted the puck before moving in on Hellebuyck, toe dragging to get a better shooting angle and beating him high glove side.
Adam Lowry scored at 17:51, knocking a cross ice backhand pass from Mathieu Perreault past Price to make it 3-1.
The goal was his 2nd of the playoffs.
With Pierre-Luc Dubois off for cross checking, Nick Suzuki got his stick down and scored from just outside the crease off a bullet pass from rookie Cole Caufield who was along the goal line to give Montreal a 4-1 lead.
Armia scored into an empty net at 16:42 of the third to make it 5-1 Montreal.
Game four goes Monday night in Montreal.
SHOTS-WINNIPEG 27 MONTREAL 33