Montreal needed one point from the game to win the division because it was two ahead of the second-place Tampa Bay Lightning entering Saturday, the final day of the NHL regular season.
The Canadiens will face the Ottawa Senators in the Eastern Conference First Round of the Stanley Cup Playoffs, which begin on Wednesday.
"First of all, it's been a successful season for us," Canadiens defenseman P.K. Subban said. "Getting 110 points and 50 wins is a big deal for our group, and I think it is well-deserved. We played hard all year, and now it's great to know we're going to focus on the playoffs.
"We haven't put a lot of time into thinking about who we're going to play, but we know we have to go through everyone to win a Stanley Cup."
Phil Kessel put the Maple Leafs ahead 1-0 at 3:58 of the first period when he took a pass from center Tyler Bozak and completed a wraparound by stuffing the puck behind Canadiens goaltender Carey Price. Kessel ended an 11-game goal drought with his 25th of the season.
The Canadiens tied it 16 seconds later after Maple Leafs goaltender Jonathan Bernier slipped behind the Toronto net. Bernier accidentally passed the puck in front to center Nazem Kadri, who wasn't expecting the pass and was unable to control the puck. It deflected to Desharnais, who scored his 14th goal.
It was the 11th time this season the Maple Leafs have given up a goal less than one minute after they scored.
Casey Bailey restored Toronto's lead at 14:04 of the opening period when he tipped a Dion Phaneuf snap shot from the point high into the net past Price for his first NHL goal. The assist was Phaneuf's 400th NHL point.
The Canadiens started the second period by killing off the final 39 seconds of a hooking penalty assessed to Galchenyuk late in the first period. They tied it when Tomas Plekanec converted a 4-on-1 by snapping a shot past Bernier for his 26th goal at 00:59.
Toronto again took the lead at 5:11 when James van Riemsdyk sent a backhand pass through his legs from behind the net to a streaking Bozak, who fired his 23rd goal into the far side of the net.
Bernier was caught wandering late in the second period, and it cost the Maple Leafs again. He went behind the net to retrieve a loose puck and attempted a pass to defenseman Jake Gardiner, but the puck deflected to Montreal's Jacob De La Rose, who shot it into the empty net to make it 3-3 at 19:51.
The Canadiens appeared to take the lead when Brendan Gallagher scored from a scramble in front of the Maple Leafs net at 8:45 with Toronto defenseman Jake Gardiner off for hooking, but the goal was disallowed because of goalie interference.
Price made saves against Kadri and Kessel on back-to-back breakaways midway through the third. Bernier stopped Montreal's Lars Eller on a breakaway in overtime.
The Maple Leafs, who have missed the playoffs nine of the past 10 seasons, approached their final seven games of the regular season as a best-of-7 series. They lost four of the seven.
Toronto coach Peter Horachek, who took over from Randy Carlyle midway through the season, said he tried to instill a system that would be successful in the second half, but the Maple Leafs did not buy in.
"We didn't lose most of our games because of a lack of offense; we lost because of a lack of defense," Horachek said. "Individuals resist change. People don't want to change. They want to keep it in the comfort levels where they are."
"There has to be a new approach to next year," Horachek added. "There has to be a commitment to being better conditioned. There has to be a new attitude and approach to how we want to do things. This has to change.
"Ultimately this team had to change. If we continued to do what we were doing in November, we might have been fighting to make the playoffs, but we weren't going to win. Is that what we want? We want to build something to win a Stanley Cup. That is the change we have to make; not just be competitive to make the playoffs. We have to have higher expectations."
The Canadiens, who have Stanley Cup aspirations, will face a hot team in the Senators and a very hot goaltender in Andrew Hammond.
"I don't think much about that," Subban said. "I just go out and play. It's the playoffs -- the most fun time of the year. I don't really care who we're playing. It's not about that. It's about how we're playing."
Gallagher admitted the Senators are going to be a formidable opponent in the first round.
"I think we're excited," Gallagher said. "I think it's going to be a challenge. We know how well they're playing lately and they have a ton of confidence, but our team is the same way. We have a ton of confidence too. When the puck drops, we just need to play our style and we're confident in ourselves."