Jeff Petry scored on a breakaway 52 seconds into overtime after Montreal blew a 3-0 lead.
Carolina's Justin Williams tied the game 3-3 with a redirection with 1:17 remaining in the third period and backup goalie Alex Nedeljkovic pulled for an extra attacker.
[WATCH: All Hurricanes vs. Canadiens highlights]
"It was a big goal by Jeff," Canadiens forward Brendan Gallagher said. "We've got to do something about 6-on-5. Obviously too many goals are ending up in our net with the extra guy on the ice."
Video: EA Sports Overtime Winner: Petry wins it for Montreal
Montreal goalie Charlie Lindgren made the save on Andrei Svechnikov from in close with one second remaining in the third.
"I had to find a way to make a save there, and thankfully I did," Lindgren said.
Hayden Fleury got the Hurricanes within 3-1 when he scored at 7:45 of the second, and Joel Edmundson scored at 1:14 of the third to make it 3-2.
Video: CAR@MTL: Fleury nets wrister through screen in front
"We can't chase games," Williams said. "Chase, and chase and chase; it's exhausting. Our urgency from the start of the game has to be better. It's blatantly honest and it's pretty obvious."
Carolina (35-24-5) trails the Columbus Blue Jackets by one point for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
"We've done it all year, coming back, but it's just why we weren't really ready from the start tonight was a little frustrating," Hurricanes coach Rod Brind'Amour said.
Carolina forward Sebastian Aho was held without a point for the first time in 15 games. He had 21 points (12 goals, nine assists) during his 14-game streak, which was the third longest in the NHL this season (Jack Eichel, Buffalo Sabres, 17 games; Patrick Kane, Chicago Blackhawks, 15 games).
"You're not going to play great every night," Brind'Amour said of Aho. "And give their team credit, they were on it at the start, and that had a lot to do with making him not be a factor. But he's world class, he means the world to us."
Phillip Danault and Brendan Gallagher each had a goal and an assist for the Canadiens (30-28-9), and linemate Tomas Tatar had three assists.
Video: CAR@MTL: Gallagher hammers slap shot past Forsberg
Lindgren made 27 saves in his first start since Feb. 4 for Montreal, which won for the first time in three games and is 3-1-1 in its past five.
"There's certain practices we say, 'Boy, was he good today,'" Canadiens coach Claude Julien said. "That's what happens when you don't play that much, your practices become your games. And if you practice well, then usually it pays off, and it paid off for him tonight. I thought he played a solid game."
Anton Forsberg made 17 saves in his second straight start for Carolina before he was pulled at 4:03 of the second period after Gallagher scored Montreal's second goal in 17 seconds to take a 3-0 lead.
Nedeljkovic made 18 saves in relief. He allowed four goals on 16 shots before he was pulled in a 4-1 loss to the Dallas Stars in his season debut Tuesday.
"I think [Forsberg] played good," Edmundson said. "It's just one of those things where you switch it up and hopefully spark our team, and I thought that [Nedeljkovkic] did a great job when he came in."
Forsberg made 29 saves in his Hurricanes debut Friday, a 3-2 loss to the Colorado Avalanche.
"They're very capable goalies," Williams said. "It could easily have been 4-0, then we'd be out of it. So it was an important point, and we dug in the last half of the game."
Carolina goalies Petr Mrazek (concussion protocol) and James Reimer (lower body) were injured in a 6-3 win at the Toronto Maple Leafs on Feb. 22, when emergency backup David Ayres made eight saves on 10 shots.
Danault gave Montreal a 1-0 lead at 2:40 of the first when he cut across the goal mouth to deflect Tatar's pass.
Video: CAR@MTL: Danault taps Tatar's feed home in front
Max Domi made it 2-0 when he scored at 3:46 of the second to extend his point streak to five games (four goals, three assists).
They said it
"The past couple of games, it's taken us until the third period, and this time of year, it's unacceptable. So we've got a few days off here to kind of regroup and come back fresh, but we've definitely got to figure out our starts." -- Hurricanes defenseman Joel Edmundson
"They were desperate and they were throwing pucks at the net and were able to get that second one from that point shot with the screen, and then I guess the high tip. So we've got to just find a way to close these games out without giving up goals late." -- Canadiens defenseman Jeff Petry
Need to know
Aho's point streak was the third longest in Hurricanes/Hartford Whalers history. Mike Rogers had a 17-game point streak in 1979-80 and an 18-game point streak in 1980-81 for Hartford. ... Tatar set a new NHL career high for points with his 59th when he got his first assist on Danault's goal. He has 61 points (22 goals, 39 assists) in 66 games after he had 58 in 80 games last season. ... Canadiens goalie Carey Price had started the previous 11 games and 17 of 18.
Hurricanes: At the Philadelphia Flyers on Thursday (7 p.m. ET; NBCSN, NBCSP, NHL.TV)
Canadiens: At the New York Islanders on Tuesday (Barclays Center; 7 p.m. ET; MSG+, TSN2, RDS, NHL.TV)
Video: Petry's OT winner propels Canadiens past Hurricanes