After nine rounds of a tiebreaker couldn't produce a winner, Ondrej Pavelec stopped Toronto's Nikolai Kulemin to lead off the 10th frame.
Then Bogosian finally ended it when he came in on Reimer, hesitated, and banged a shot off the crossbar that ended up behind the Maple Leafs goalie for the game-winner.
Blake Wheeler put in two regulation goals while James Wright and Antti Miettinen also scored for the Jets, who improved to 5-0-1 in their last six games and leapfrogged Carolina for the top spot in the division.
Pavelec finished with 28 saves for Winnipeg.
Joffrey Lupul celebrated his return from a 25-game absence with a fractured right forearm with a pair of goals for the slumping Leafs, who tumbled to their fifth straight loss.
Ben Scrivens started in net but was pulled in the second period after surrendering four goals in 26 shots.
Reimer stopped 7-of-7 the rest of the way.