Kovalev scored in the fourth round of the shootout after Montreal's Saku Koivu and Edmonton's Ales Hemsky were the only scorers in the first three round. Rookie Guillaume Latendresse gave Montreal a 2-1 lead 1:04 into the third period with his first NHL goal, but Petr Sykora tied it for Edmonton at 15:30 to force overtime.
Brad Winchester got his first career goal in the first period for the Oilers (7-6-1), who had lost four of their previous five games in regulation.
Craig Rivet scored in the second frame for Montreal (9-3-3), which was coming off a 2-1 loss to New Jersey. The Canadiens are 2-3 in shootouts, while it was a first for the Oilers this season.
Montreal looked to take control when Michael Ryder fed Koivu, who tried to jam to puck under Dwayne Roloson and the puck lay on the goal line for Latendresse to tip in.
The sellout crowd of 21,273 rose in a standing ovation and chanted Gui Gui, even if the goal was originally awarded to Koivu and only a moment later given to the popular 19-year-old Latendresse.
Latendresse was promoted to the Koivu line when team scoring leader Chris Higgins suffered a sprained ankle against the Devils.
Oilers veteran Ryan Smyth won a battle in the corner and fed Sykora for a high wrist shot from the right circle to tie the game.
The audience included NHL commissioner Gary Bettman, fresh from general managers' meetings in Toronto where the main topic was inter-conference play and why teams like Montreal and Edmonton only play each other once per year, if at all.
They looked like bitter rivals as three fights broke out in the second period, with Mike Komisarek facing Winchester, Aaron Downey battling Jason Smith and Sheldon Souray fighting Matt Greene after the Oiler defenceman's knee-on-knee hit on Canadiens winger Mike Johnson.
Johnson was helped off the ice favouring his right knee, but returned for the third period.
The opening period was all Oilers, but goaltender David Aebischer was solid until Winchester redirected Marty Reasoner's pass on a rush between the pads on Edmonton's 15th shot at 13:54.
A Montreal power play had just ended when Rivet put a wrist shot in under the crossbar 6:32 into the middle period.
Smith went down when he was struck on the side of the face by a Mark Streit point shot while killing a penalty in the third period, but wasn't cut and got a warm ovation as he got up and skated off.
Aebischer, a former Colorado Avalanche, is 9-3 in his career against Edmonton.
Notes - Montreal D Mathieu Dandenault, who missed a 10th game with a leg injury, is to resume skating this week and may return next week. . . Higgins will miss six weeks with a sprained ankle. . . F Patrick Thoreson and D Daniel Tjarnqvist were healthy scratches for Edmonton.