Trailing 3-1 in the third period, Koivu scored twice and added another in a shootout as the Canadiens defeated the Maple Leafs 4-3 despite being outskated for most of regulation time and overtime and being outshot 39-22.
"We got some hope after our second goal," said Koivu. "It just seemed we had more energy."
Defenceman Sheldon Souray, known for his cannon point shot, got the game-winner in the shootout after the two captains - Koivu and Toronto's Mats Sundin - exchanged goals.
Souray, in his first career shootout attempt, faked a slapshot and beat Andrew Raycroft with a move to the forehand.
"I thought he was going to do something ridiculous - just bomb it away," said Raycroft. "Guys have done it this year (like Minnesota's Brian Rolston) and it's worked.
"But I should have moved a little more and had a backup plan."
Souray confirmed he was thinking of blasting it from the slot.
"If I had room, I definitely would have shot it, but I think I managed to freeze him a bit," said Souray, who was surprised coach Guy Carbonneau picked him to shoot.
"The pressure wasn't on me. I don't think anyone expected me to score."
Toronto's Kyle Wellwood had the final shot and put it over the Montreal net. Toronto's Tomas Kaberle and Montreal's Alex Kovalev missed on earlier attempts.
Wellwood, Sundin and Ian White scored in regulation time for slumping Toronto (13-10-5), which earned a point but is winless in four games.
Toronto also remained without a win since Sundin returned from an elbow injury four games ago, although this was one they would have won had Huet not been outstanding in goal.
"We almost doubled their shots and we ended up losing," said Sundin, who had eight shots on goal. "I thought we did a lot of good things.
"But sometimes when you're struggling, the bounces don't go your way. Montreal found a way to win even though I thought we were the better team."
Toronto coach Paul Maurice accused the Bell Centre Zamboni driver of leaving rough ice in the end the Leafs shot from in the shootout. He said they did the same when Toronto visited on Oct. 28, although the Leafs won that time.
"I asked the refs to bring the Zamboni back out," said Maurice. "It was beautiful because they left a six-inch swath, so our chute was narrower. But anyway, that didn't cost us the game."
After Wellwood gave Toronto a 3-1 led with a power play goal 2:21 into the third period, Koivu scored twice in a 5:07 span to tie the game with 4:36 left to play to force overtime.
The Canadiens (15-7-3), who were beaten 4-2 in Carolina on Thursday, have not lost consecutive games this season.
Alexander Perezhogin had the other goal for Montreal.
The Bell Centre sellout crowd of 21,273 had its usual competing chants from fans of both clubs when the Leafs visit and they saw Toronto play a solid skating game.
But after Wellwood's goal, Koivu had the building in a roar as he took the puck from a faceoff and jammed it in to cut the lead to one with 8:43 left to play.
After White was sent off for shooting the puck over the glass, Koivu redirected a feed from Kovalev to the top corner to tie the game and force overtime.
The Canadiens didn't skate in the first period, a bad idea against a hungry opponent, and the Leafs scored twice.
No one picked up trailer White on a rush and the Leafs defenceman was in the slot to fire a Michael Peca pass into an open side 9:56 into the game.
Sundin made it 2-0 when he blew past Craig Rivet on the left side and lifted the puck over Cristobal Huet from in close at 12:09.
Montreal got one back 9:19 into the second frame when Nik Antropov and Brendan Bell collided at the Montreal blue-line, sending the Canadiens in three-on-one the other way. Perezhogin beat Andrew Raycroft after taking a drop pass from Guillaume Latendresse.
In the final minute of the second period, Kovalev was slapped for an extra minor for arguing a hooking call and Toronto used the fresh ice to start the third to bang Bryan McCabe's rebound into an open side.
Carbonneau said there was "no excuse" for Kovalev's extra penalty. "It was 2-1 and we were trying to come back."
Montreal lost Rivet late in the second period after he was struck on the right wrist or arm by a wrist shot and did not return. Carbonneau said the team would learn how serious the injury is on Sunday or Monday.
Near the end of the first period, Toronto's Darcy Tucker took offence to a hit from Francis Bouillon and, after a charge from behind and some shoving, the Canadiens defenceman floored him with a straight right. Tucker got up with a bloody nose but stayed in the game.
Toronto returns home to play Atlanta on Tuesday night.
The Canadiens play host to Boston on Monday night and will salute the 82-year rivalry between the clubs in a pre-game ceremony, with former stars from both clubs like Elmer Lach, Jean Beliveau, John Bucyk and Ray Bourque taking part.
Notes - Wade Belak and Aleksander Suglobov were healthy scratches for Toronto as D Brendan Bell returned to the lineup after sitting out six games. . . Montreal reunited last year's solid defence pair of Mathieu Dandenault and Francis Bouillon and sat out D Mark Streit and D Janne Niinimaa. F Chris Higgins has now missed 12 games with a sprained ankle.