The Rangers, who lead the best-of-7 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.
"I think you're starting to see us playing Montreal Canadiens hockey," Montreal forward Max Pacioretty said. "I don't think you've really seen it in this series just yet.
"It was great to see a little taste of it. I think we still have more. I think we still have little things to work on. And it should be a fun one going back there."
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 just a strange game," Rangers coach Alain Vigneault said. "It was a different game from what we've seen so far in the series, but now we're going home."
It was the third time in these playoffs that the Canadiens have faced elimination.
As they did after falling behind 3-2 to Boston in the conference semifinals, they came up with their best hockey to stay alive.
Pacioretty feels that being under pressure helps his team focus.
"There's a lot of distractions and noise in this city and I think that when we let it affect us and we don't worry about ourselves and what we can control we get away from our game a bit," he said.
"Now that we're in desperation mode and our backs are against the wall, we're just worrying about what we can control.
"When we get four lines buzzing like that and play the way we're capable of we have a lot of success. Hopefully we can do that next game."
The Rangers weren't facing the same desperation and spent the game trying to play catch-up. Going 1 for 7 on the power play didn't help.
Neither did getting an ordinary game from the often extraordinary Lundqvist.
He let in four goals on only 19 shots before he was pulled at 8:58 of the second period in favor of Cam Talbot.
"I pulled him because I thought at that time we needed a little momentum shift, and I thought it might catch everybody's attention," Vigneault said. "It did for a while. Obviously it didn't work out."
Montreal's Dustin Tokarski, starting a fourth game since Carey Price was injured in the series opener, allowed four on his first 14 shots, but then made some big saves in the third to preserve the win.
It prompted a bold observation from Bourque.
"Everybody talks about how [Lundqvist] is a great goalie: Has he been better than [Tokarski] this series?" Bourque asked. "I don't think so.
"[Tokarski] made some big saves for us, too. We had a couple bad bounces but our power play was the difference. We got some traction, got a couple goals in tight."
Bourque's second goal proved to be the winner and came just after the Rangers came back to tie the game at 4-4 in the second period.
"Everybody was ready for this game," Bourque said. "We knew the situation.
"It was just a see-saw battle back and forth. [Dale] Weise made a great play to me. I called for the puck and somehow it got through. It was nice to get in there."
Bourque has been a different player in the postseason than he was while scoring only nine goals in 63 games in the regular season.
"It's easy to sit back and get down on yourself after giving up that lead, but coming out right away and putting that in the top corner, that's the difference in the game," Pacioretty said. "It was a huge boost for us."
Weise was wobbly when he got up and went for treatment, but returned to the bench late in the period.
The NHL player safety department announced that Moore will have a hearing on Wednesday to see if further discipline is in order.
"The league will do what it has to do," Vigneault said of the hit. "John is not the type of person who would try to hurt someone, but it was a late hit."
There was no mention of Bourque, who took a slashing major at the end of the game.
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 Tokarski's stick midway through the first period.
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 fired up the crowd and the team.
Only 22 seconds in, 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.
Stepan made it 1-1 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.
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.
Desharnais scored into an empty net during a New York power play at 15:43.
Asked if he had been head-butted, Weaver said: "Ah, so many things happened I don't really know. I was more concentrating on where the puck was. I wasn't really worried about what he was doing."
The Canadiens scored seven goals in a playoff game for the first time since beating the Hartford Whalers 7-4 on April 27, 1992.
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.