Kovalchuk scored on a rebound after Maple Leafs goalie Jack Campbell (28 saves) made the initial save on Nick Suzuki's backhand shot.
Marco Scandella tied the game with 2:33 left in the third period, and Carey Price made 22 saves for Montreal (27-23-7), which has won three in a row. The Canadiens are 9-3-0 since going eight straight without a win (0-7-1).
"I think right now, of course we've got to rely on ourselves, but we've got to be aware of what's going on around the League because every game, every team, it's so close right now," Kovalchuk said. "And we just believe in ourselves. We battled through some injuries, some flu and everything, and I think it's just going to make us stronger."
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Campbell made 26 saves in his Maple Leafs debut Friday, a 5-4 overtime win against the Anaheim Ducks, two days after he and forward Kyle Clifford were acquired in a trade with the Los Angeles Kings.
He lost his shutout Saturday when Scandella scored on a slap shot from the left point to tie it 1-1.
"I'm just definitely not happy, giving that one up, you know, to give them life again and blow that lead and in the third," Campbell said. "I just know my ability and I'm confident with my team in front of me, and they battled all night, so one more save and we get two points."