It was the fourth time in Rangers history that they won a game after trailing by at least four goals, first since the 1991-92 season.
After New York tied the game 5-5 on Brendan Lemieux's second goal of the game at 8:40 of the third, Jacob Trouba put the Rangers ahead 6-5 at 12:10.
[WATCH: All Rangers vs. Canadiens highlights]
Trailing 4-0, Filip Chytil, Pavel Buchnevich and Lemieux scored in a span of 3:20 to draw New York within 4-3.
Artturi Lehkonen scored his second goal of the game at 5:20 of the third to put Montreal up 5-3. Artemi Panarin scored 31 seconds later to make it 5-4 at 5:51.
Lemieux scored shorthanded off a pass from Brett Howden on a 2-on-1 to tie it.
Video: NYR@MTL: Lemieux scores a SHG on an odd-man rush
"I'm sure this term gets used by more than one person, but I know Patrick Roy said it to my dad (Claude Lemieux), and they tried to use it throughout their careers when they were winning, and it's, 'Never too high, never too low,'" Lemieux said. "And I think that that was one of those things. You know, we were playing well, and the score didn't reflect the way we were playing.
"So we just had to stay with it, not too high, not too low, not down because the score's down, or not up because the score is up, and I think tonight we emulated that."
Alexandar Georgiev made 38 saves for the Rangers (10-9-2), who lost 4-1 at the Ottawa Senators on Friday.
"It's just an unreal feeling that the guys kept at it, nobody gave up," Georgiev said. "It's rare that you get to come back from 4-0. It's a huge mental win."
Lehkonen and Max Domi each had two goals and an assist, Shea Weber scored, and Carey Price made 28 saves for Montreal (11-7-5), which has not won in its past four games (0-2-2).
"It's different than those three other games," Canadiens coach Claude Julien said. "You've got a 4-0 lead, you finally are scoring goals, but then you get sloppy. And I told the guys after the first period we were up 3-0 for one reason only, we capitalized on our chances, they didn't, because we didn't dominate the first period. [The Rangers] came to play tonight."
Video: NYR@MTL: Lehkonen snaps home wrister from the slot
The six goals-against were a season high for Montreal.
Domi scored his first goal in 10 games at 2:03 of the first period to give the Canadiens a 1-0 lead.
Lehkonen made it 2-0 at 10:00, and Domi put the Canadiens up 3-0 with his second goal of the period with 34 seconds remaining.
Weber scored on a delayed penalty at 2:51 of the second to make it 4-0.
The Rangers scored two goals in 56 seconds to cut the lead in half. Chytil scored at 6:10 to make it 4-1, and Buchnevich made it 4-2 at 7:06 when he cut across the goal mouth and had the puck hit defenseman Cale Fleury's skate and slide beneath Price.
Lemieux got New York within 4-3 when he tipped Adam Fox's shot from the point at 9:30.
"We were still up," Price said. "We were pretty confident going into the third period. We just wanted to stay on the gas, and I think they came out motivated as well. That's a tough loss for us, for sure."
Video: NYR@MTL: Trouba nets shot off defender
They said it
"That's what we've been looking for all year, is that resiliency. I mean, that happens sometimes in hockey, right? But for once, we didn't get away from what we needed to do and we were rewarded." -- Rangers forward Chris Kreider
"You know that's on us as a group, nothing else you can point your finger at. It can't happen in this league. Too many guys, even when we had the lead, and I definitely put myself in that category, weren't good enough, and we had too many passengers tonight. We had a few guys going, but a lot of us, for whatever reason, weren't good enough." -- Canadiens forward Brendan Gallagher
Need to know
The Rangers won 8-6 after trailing 6-1 against the Washington Capitals on Dec. 26, 1991, and rallied from down 4-0 in a 5-4 overtime victory against the New York Islanders on Nov. 9, 1984, and an 8-5 win against the Vancouver Canucks on Nov. 21, 1986. ... There have been five games in the NHL this season when a team has won after trailing by four goals; the Florida Panthers have done it twice. ... New York matched its season high for goals (6-4 win against the Winnipeg Jets on Oct. 3, 6-2 win against the Buffalo Sabres on Oct. 24). ... It was Lemieux's third multigoal game in the NHL and first this season. ... Domi and Lehkonen each had their first multigoal game of the season.
Rangers: Host the Minnesota Wild on Monday (7 p.m. ET; ESPN+, MSG, FS-N+, NHL.TV)
Canadiens: Host the Boston Bruins on Tuesday (7 p.m. ET; ESPN+, TSN2, RDS, NESN, NHL.TV)
Video: Rangers fight back to defeat Canadiens, 6-5