Bellemare beat Canadian goalie James Reimer high to the stick side, prompting a wild celebration with his teammates at centre ice.
It was just the second French win ever over Canada at the World Championship, dating back to 1931.
Stephane Da Costa notched two power play goals for France. Brayden Schenn and Erik Gudbranson scored for Canada.
Few expected this game to be so close, but given that France stunned the then-defending champion Russia 2-1 last year, it's obviously not wise to underestimate coach Dave Henderson's squad.
French goalie Cristobal Huet was magnificent, denying Kyle Turris, Sean Monahan, and Matt Read in the shootout. Huet was the first Frenchman to win the Stanley Cup in 2010 as a Chicago Blackhawks backup. Canada outshot France 36-29.
Canada, the two-time defending Olympic champion, is seeking its first IIHF World Championship gold medal since 2007 and first medal of any shade since 2009.
France, which sits 12th in the IIHF World Ranking, is looking to improve on last year’s 13th-place finish.
There was a good, aggressive tempo, especially considering that it was the opener for both sides here in Minsk. The crowd largely supported France, with chants of “Shaibu!” supplementing cries of “Allez Les Bleus!”
The Canadians didn't record their first shot on goal for four and a half minutes. But they peppered Huet during their first power play after French captain Laurent Meunier hauled down his Canadian counterpart, Kevin Bieksa.
Stephane da Costa drew first blood for France on the man advantage with 2:57 left in the opening frame. He fired home a rebound from the goal line to Reimer’s right after Yohann Auvitu’s shot from the hash marks was stymied.
Canada quickly equalized on the power play before the first buzzer. At 19:46, Cody Hodgson sent a nice pass from the left-side boards to Schenn, who zinged a wrister over Huet’s glove.
Huet made a magnificent glove save on Morgan Rielly during a mid-second period French power play, diving cross-crease to foil the high attempt from the right faceoff circle.
Late in the second period, Canada ran into a stretch of penalty trouble, including a pair of slashing minors to Troy Brouwer. But France couldn’t capitalize even with two 5-on-3 advantages.
Gudbranson gave Canada a 2-1 lead just past the halfway mark of the third period. From the hash marks, he snapped a shot off Huet's right post, and it bounced off the netminder and just over the line. Nathan MacKinnon, considered the NHL's front-runner for rookie of the year with the Colorado Avalanche, earned his first World Championship point with the assist.
With 7:25 left and France on the power play, Da Costa tied it up when he was left alone in front of the net and Antoine Roussel centred the puck from behind the goal line. Da Costa deked Reimer on the backhand, depositing the puck into the gaping cage.
Huet made another great stop on Hodgson just before the end of the third.
Overtime solved nothing. The best chance went to France's Nicolas Besch on a partial break, with Reimer making a nice pad save.
It was the tenth all-time World Championship meeting between these nations. Canada won eight of those encounters, with France’s lone previous victory coming at the 1995 tournament in Sweden (4-1 on April 25).
The French wore “2017” on their helmets and shoulders, promoting their co-hosting of the 2017 IIHF Ice Hockey World Championship with Germany.
Next up for Canada is Slovakia on Saturday, while France takes on Italy on Sunday.