Bobrovsky made 35 saves, and kept his team hanging around in the third period long enough to erase a two-goal deficit en route to a 3-2 overtime win.
Toronto looked to have secured two points, but Columbus scored twice in the final 4:35 of regulation and Artemi Panarin completed the comeback at 3:11 of the extra period.
"Those kind of wins make good atmosphere in the locker-room, lots of energy, lots of happiness," said the 2017 Vezina Trophy winner.
"That’s a good win for us as far as going through some of the crap we’ve been going through," said Blue Jackets coach John Tortorella. "I really respect how they kept their head up and kept playing."
The Leafs (25-16-3) entered the third period up two goals on Columbus (25-16-3) and did everything they could to try and put the game out of reach.
"Some nights when you’re ahead you get on your heels and the other team takes it to you. That wasn’t the case at all. It was the opposite way around. But they scored on their opportunities that we gave them," said Leafs coach Mike Babcock.
"That’s a free point we gave away."
Nick Foligno started the comeback with 4:55 remaining on a play that Frederik Andersen lost the bouncing puck on before it found its way behind him and in the net. Jordan Schroeder was originally credited with the goal before it was changed to the Columbus captain.
"We knew if we scored one somehow that it changes momentum," said Tortorella. "The saves (Andersen) made during the game and that one goes in… it’s a funny game."
Rookie Pierre-Luc Dubois then tied it with 2:47 left on the clock, snapping a wrist shot blocker side on Andersen from the slot after defenceman Morgan Rielly turned the puck over on a long outlet pass.
"Line change where we just could have been patient, we were under no pressure and threw it out and they came in on an odd-man rush and in the end that cost us the hockey game. It’s all part of the learning process," said Babcock.
"I think Morgan’s having a real good year, made a mistake tonight."
James van Riemsdyk and William Nylander scored for the Leafs while Andersen stopped 30 shots in defeat.
"At least 50 minutes or so we played great, I think we could have put it away in the third and we didn’t. Then they get a good bounce and, yeah, it’s tied up," said Andersen.
Andersen held the Leafs in it long enough for van Riemsdyk to open the scoring at 11:09 of the second period, deflecting a Roman Polak point shot past Bobrovsky for his 18th goal of the season.
The Leafs maintained momentum and Nylander doubled the lead on the power play with 3:58 to go in the second, rifling a wrist shot over the glove hand of a screened Bobrovsky for his ninth of the campaign.
Blue Jackets forward Sonny Milano left the game in the second period with an upper-body injury and didn’t return after being on the receiving end of a check from Andreas Borgman. Tortorella didn’t have an update post-game.
The 18,933 in attendance at Air Canada Centre were chanting "Freddie, Freddie," for Andersen, who stopped all 25 shots his way through 40 minutes. But they were left silent as Andersen’s near shutout quickly turned into defeat.
"I think when the overtime really gets going is when there’s chances each way and unfortunately, I wasn’t able to capitalize on mine and they came back and got one there," said Tyler Bozak, who was turned away on a breakaway in overtime by Bobrovsky.
"It’s a tough way to lose but we’ve got to look forward to the next game."
Toronto hosts Ottawa Wednesday before starting its bye week. Columbus’ next game is Thursday in Buffalo.
Toronto defenceman Travis Dermott played his second straight game since being called up from the AHL Marlies. The 34th pick from the 2015 NHL Draft started on a pairing with Borgman but quickly found himself alongside Jake Gardiner.