Sheldon Souray scored the winner after Koivu and Leafs captain Mats Sundin exchanged shootout goals. Souray faked a slapshot, moved to the forehand and beat Leafs goaltender Andrew Raycroft.
Toronto's Kyle Wellwood had the final shot and put it over the Montreal net.
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 and the shootout.
Sundin and Ian White also scored for slumping Toronto (13-10-5), which is winless in four straight games.
The Canadiens (15-7-3), 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 of the storied rivals.
Toronto played a solid skating game and outshot Montreal 36-21 in regulation and 39-22 overall.
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 Alex Kovalev to the top corner to tie the game and force overtime.
Montreal lost defenceman Craig Rivet late in the second period after he was struck on the right wrist or arm by a wrist shot and did not return.
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 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 Raycroft after taking a drop pass from Guillaume Latendresse.
Near the end of the first period, Toronto's Darcy Tucker took offence to a hit from Francis Bouillon and, after a charge 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.