Ben Chiarot scored on a pass from Max Domi on a 2-on-1 at 1:11 of overtime.
Primeau, who is 20 years, 122 days old, is the second goalie in as many seasons to record his first NHL win before his 21st birthday. Carter Hart was 20 years, 127 days old when he got his first win with the Philadelphia Flyers on Dec. 18, 2018.
[WATCH: All Senators vs. Canadiens highlights]
"It's something you dream of, and just being able to get the first one with this group of guys it's been unbelievable," said Primeau, who made his second start since being recalled from Laval of the AHL on Dec. 2. "The guys have been amazing, so it's definitely emotional and special."
Video: OTT@MTL: Tatar has his slap shot pinball home
While the Canadiens mobbed Chiarot, defenseman Shea Weber, Montreal's captain, made a point of congratulating Primeau.
"I feel like the goalie always gets left out no matter what," Weber said. "Obviously it's his first win so it's even more special, but I tend to go to the goalie first. I just feel like the swarm all goes to one guy and the goalie's just kind of the short end of the stick. So I try and make him feel a little bit special as well."
Nick Cousins and Tomas Tatar also scored for Montreal (15-11-6), which has won three straight and is 4-1-0 in its past five games following an 0-5-3 skid.
"I knew once we once we got a couple wins under our belt, we'd get that good feeling back and get rolling in the right direction, and I think that's what you're seeing," Chiarot said. "We're playing the right way, not giving up a lot of big fat chances like we were when we were losing all those games. We've tightened up a lot, given up a lot less, and now it's paying off. We look good out there."
Brady Tkachuk and Connor Brown scored in the third period, and Anders Nilsson made 26 saves for Ottawa (13-17-2), which is 2-6-1 in its past nine games.
"We battled back and we played well for, I'd say, the second half of that game," Tkachuk said. "They took it to us a little bit in the first."
Video: OTT@MTL: Primeau gobbles up Pageau's backhander
Cousins gave Montreal a 1-0 lead with a power-play goal at 13:36 of the first period. He deflected Domi's shot from the right side for his fourth goal.
"It was a nice play," Cousins said. "We kind of talked about that before the game and it just happened to work out, all I had to do was put my stick there."
Tatar made it 2-0 when he scored his 11th goal 29 seconds into the third. His wrist shot from the slot appeared to go in off Ottawa defenseman Mark Borowiecki's arm.
Tkachuk drew the Senators within 2-1 on a power play when he scored his 11th goal on a shot from the slot at 4:30.
Brown tied it at 2-2 with a backhand shot at 13:05 of the third period.
"We knew if we just kept getting pucks on net there was going to be a bounce that goes in," Tkachuk said. "So we were just really focused on getting pucks. And Primeau, it was his second game so I've been fortunate enough to play with him and he's a great goalie, but I just knew if we kept getting pucks on him there would be some chances to score."
Video: EA Sports Overtime Winner: Chiarot wins it for Habs
They said it
"I thought our guys played hard, you know, to be down 2-0 in Montreal, we didn't stop and I thought we kept going. You know, you lose in overtime, it is what it is, but I thought our guys played really hard." -- Senators coach D.J. Smith
"That eight-game [winless] streak wasn't fun. I mean, it wasn't fun coming to the rink, you know it [stinks] when you're losing like that. And I mean at the end of the day, when things turn around you kind of look back at that stretch there and it brings us closer together. It makes us mentally stronger and right now things are clicking, things are going well. We've just got to keep going." -- Canadiens forward Nick Cousins
Video: OTT@MTL: Cousins deflects home Domi's wrister
Need to know
Senators defenseman Ron Hainsey became the third player chosen in the 2000 NHL Draft to play 1,100 games in the League along with Scott Hartnell (1,249) and Justin Williams (1,244). … Primeau is the fifth-youngest goalie to get his first NHL win with Montreal: Len Broderick (19 years, 19 days; Oct. 30, 1957 at the Toronto Maple Leafs), Patrick Roy (19 years, 141 days; Feb. 23, 1985 against the Winnipeg Jets), Jose Theodore (20 years, 54 days; Nov. 6, 1996 at the Mighty Ducks of Anaheim), and Carey Price (20 years, 55 days; Oct. 10, 2007 at the Pittsburgh Penguins).
Senators: Host the Columbus Blue Jackets on Saturday (1 p.m. ET; SN, TVAS, FS-O, NHL.TV)
Canadiens: Host the Detroit Red Wings on Saturday (7 p.m. ET; SN360, CITY, SNE, TVAS, NHLN, FS-D, NHL.TV)
Video: Chiarot buries OT winner as Primeau earns first win