The Rangers, who lead the best-of-seven series 3-2, will have another chance to earn a trip to the Stanley Cup final in Game 6 on Thursday night in New York.
Alex Galchenyuk, Tomas Plekanec and Max Pacioretty also scored for Montreal, which outshot the Rangers 28-27.
"That was a lot of fun," Pacioretty said. "We came out strong. That's a group effort. Everyone contributed tonight and that's makes it more special."
Derek Stepan, playing with a guard on his helmet to protect a broken jaw suffered from a Brandon Prust hit in Game 3, returned to the lineup to score twice for the Rangers. Chris Kreider had a goal and three assists and Rick Nash also scored.
"It was a different game than what we've had so far in this series," Rangers coach Alain Vigneault said.
At 10:41 of the third, Rangers defenseman John Moore was given a major penalty and was ejected for a blindside, open ice hit on Dale Weise that was similar to Prust's hit on Stepan. Weise was wobbly when he got up and went for treatment, but returned to the bench late in the period.
"The league will do what it has to do. I think he was penalized on the ice," Vigneault said. "John is definitely not the type of player to try to hurt someone, but it was a late hit and it was the right call on the ice."
It was a night of strange bounces and spotty goaltending, even if New York's best chance of the game had Carl Hagelin's shot stopped by the end of Dustin Tokarski's stick midway through the first period.
Lundqvist was pulled in favor of Cam Talbot after allowing four goals on 19 shots.
"I pulled him because I thought at that time we needed a little momentum shift," Vigneault said. "I thought it might catch everyone's attention. It did for a while. But obviously it didn't work out."
Tokarski stayed in despite allowing four on his first 14.
The Rangers did a good job of canceling the initial rush Montreal gets from its pregame buildup in winning the opening two games of the series, but Ginette Reno's "O Canada" seemed to work this time.
Only 22 seconds into the game, Kreider was sent off for tripping and the Canadiens converted when Galchenyuk tipped in P.K. Subban's point shot from the edge of the crease at 1:48.
"We had a big power-play goal early and a few others throughout the night," Bourque said. "That was the difference for us tonight."
Stepan tied it at 10:44 of the first on a 30-foot shot off a rush that fooled Tokarski. Plekanec restored the lead with a similar goal at 12:24 as he swiped the puck between two defenders and had it beat Lundqvist.
The second period had plenty of scoring.
Pacioretty got it started on a nice pass from Brendan Gallagher at 3:44 and Bourque gave Montreal a three-goal lead when he spun and scored from close range. That chased Lundqvist.
Nash whipped a puck at the Montreal net and had it go in off defenseman Andrei Markov's skate 9:48.
Stepan got his second in a mass scramble in front of Tokarski at 12:06 and Kreider tied it on a power play at 14:12 on a tic-tac-toe play after Subban lost his stick.
The Bell Centre went quiet, but got loud again when Bourque got his second at 15:10 as he beat Talbot from 10 feet.
Several hats were thrown on the ice after Weise sent Bourque in alone to get his third of the game 6:33 into the third period.
"Weise made a great play to me," Weise said. "I called for the puck earlier when we were crisscrossing in the neutral zone and somehow it got through. It was nice to get in there and put one upstairs."
Desharnais scored into an empty net during a New York power play at 15:43.
The Canadiens scored seven goals in a playoff game for the first time since beating the Hartford Whalers 7-4 on April 27, 1992.
Notes: Canadiens defenseman Alexei Emelin, who appears to have been playing on a hurt knee, sat out with an undisclosed injury. Rookie Nathan Beaulieu played on the third pairing, while Mike Weaver moved onto the second pair with Andrei Markov. ... With eight playoff goals, Bourque is only one short of his total in 63 regular-season games. ... Prust served the second of his two-game suspension and can return for Game 6. ... With Stepan back, J.T. Miller sat for New York.
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