MONTREAL -- The Montreal Canadiens have no quit in their game, which is a good thing considering how often they fall behind.
David Desharnais scored 56 seconds into overtime Tuesday to give Montreal a 2-1 win against the Detroit Red Wings at Bell Centre.
Desharnais put a backhand of Max Pacioretty's shot over goalie Jimmy Howard from the right side of the net to give Montreal its third win in a row.
Carey Price made 27 saves for Montreal, which allowed the first goal for a sixth game in a row. The Canadiens have trailed in each of their first seven games but have won six.
"We're just relentless. I think that pretty much sums it up," Price said. "We don't feel like we're out of any hockey game so we just keep working and keep coming at you, and we feel like if we keep doing that it will pay off."
Canadiens forward Alex Galchenyuk scored on a wraparound with 3:09 remaining in the third period to tie the game 1-1.
"I thought in the third that we backed off a little too much," said Red Wings defenseman Niklas Kronwall, who played his 600th NHL game. "We didn't really go after them the way that we would like to, but in saying that (Howard) was playing strong in front of our net and it felt like we were in good shape."
Montreal won 37 of 59 faceoffs (63 percent), including 19 of 27 in the third period.
"In the end we lost too many faceoffs going down the stretch," Detroit coach Mike Babcock said. "That affected the outcome of the game."
Red Wings forward Henrik Zetterberg scored for a second game in a row, and Howard stopped 34 shots.
Galchenyuk worked free of Red Wings defenseman Kyle Quincey at the left side of the net while Montreal's Alexei Emelin shot from the left point. The left wing retrieved the rebound behind the goal and went behind to stuff the puck into the right side of the net for his third goal.
"It got tied up between (Quincey) and Galchenyuk before it went to the backboards," Howard said. "(Quincey) ended up breaking his stick and landed on my leg, then I tripped going back to the post, and he got to the puck quick and made a nice play."
Galchenyuk, who scored for a second game in a row, had an ongoing battle with Quincey after the latter was called for roughing during a heated scrum at 17:28 of the second.
"My job was to get that puck and put it on net, and his job was not letting me do it," Galchenyuk said. "I got the best of that, but the game's over now and to be honest I already forgot about it."
Pavel Datsyuk made his season debut for the Red Wings and assisted on Zetterberg's goal at 13:41 of the second period.
Datsyuk missed Detroit's first five games of the season because of a separated shoulder he sustained during training camp on Sept. 22. Normally a center, he lined up at left wing with Zetterberg handling faceoffs and Justin Abdelkader on right wing.
"Datsyuk was exactly what I expected," Babcock said. "When you miss as much time as (Datsyuk) has, you miss training camp basically, it's going to take you some time. I mean he's a talented, talented guy, but it's still going to take you time to become what you're capable of being."
Zetterberg scored his second goal of the season when he broke down the right side of the Montreal zone before cutting inside on Emelin. Zetterberg's shot went in off Canadiens defenseman P.K. Subban's skate before quickly exiting the net.
Datsyuk was denied a spectacular backhand goal at 11:51 of the third period. Referee Greg Kimmerly immediately waved his arms to deny the goal, ruling that Abdelkader had come into contact with Price when Datsyuk spun around in the slot and put a shot into the right corner.
"(Abdelkader) was just going to the front of the net and he stopped at the top of the crease and bumped into me, and I couldn't see anything," Price said. "He turned me sideways a little bit and I didn't even see Datsyuk shoot the puck, so it's the ref's discretion and I'm definitely not going to argue with him."
Babcock had a different perspective.
"What are you going to do? You're supposed to allow the goalie to play," Babcock said. "(Price) initiated the contact himself, but there was still contact."
Detroit killed Montreal's only power play and has yet to allow a power-play goal this season in 18 attempts. The Red Wings failed to score on any of their three power-play opportunities, extending their current drought to 0-for-14. They are now 2-for-24.
The Canadiens didn't allow a goal in the first period for the first time this season. Montreal, which allowed 20 goals in its first six games, has been outscored 9-3 in the first.
"We always think we've got a chance," Subban said. "I mean, that's our leadership in this room. We always know we have an opportunity to win a hockey game, and if you look at the past couple of years, in terms of how we've handled ourselves when faced with adversity, being down by goals, then scoring timely goals, teams coming back on us and us still holding on to win hockey games, I think it says a lot about our team and the character in this room."