Connor McDavid scored at 2:42 of OT for his league-leading 102nd point of the season. Linemate Leon Draisaitl set up the game-winner, his second assist of the night.
Montreal still earned a single point to secure a playoff berth in the NHL's North Division. The Canadiens (24-21-10) are winless in their last four games.
Dominik Kahun, Ryan Nugent-Hopkins and James Neal had the other goals for the Oilers (34-18-2), who extended their road winning streak to six games.
Jake Evans, Paul Byron and Artturi Lehkonen tallied for the Canadiens.
— Sportsnet (@Sportsnet) May 11, 2021
Edmonton has already locked up second place in the North Division. The Canadiens moved nine points ahead of the fifth-place Calgary Flames, who have three games in hand on Montreal.
The Canadiens will close out their regular season Wednesday against the Oilers. Edmonton will wrap up its campaign Saturday against the Canucks.
McDavid, who reached the 100-point mark on Saturday, had an assist on Kahun's goal. McDavid blew past Tomas Tatar before finding Kahun at the side of the net for the tap-in.
Evans opened the scoring with his first goal since Jan. 21, sniping a shot over Mikko Koskinen's left shoulder.
Neal answered 20 seconds later at 9:52 of the first period by flipping in a loose puck by the side of the net. He beat Canadiens netminder Jake Allen again seconds later but hit the post.
Nugent-Hopkins gave Edmonton the lead at 13:29. Kailer Yamamoto had his stick lifted on a 3-on-2 break but the puck slid to Nugent-Hopkins for his 15th goal of the season.
Byron pulled Montreal even at 5:56 of the second period. He roofed a backhand after Lehkonen's pass slid through the goalmouth.
Lehkonen beat Koskinen early in the third period but the goal was waived off after a review determined the Canadiens were offside.
The Montreal forward made no mistake at 13:04 when he wired it in from a tight angle. Edmonton outshot Montreal 35-28.
Byron returned to the lineup after missing nine games with an upper-body injury.
Forward Brendan Gallagher (thumb) and goalie Carey Price (concussion) skated Monday morning but head coach Dominique Ducharme said neither player will play before the playoffs.
Edmonton clinched its playoff spot on May 3.