Blake Coleman scored for the homeside, but Calle Jarnkrok - who was briefly a Flame late last season - would tally the winner in the third period.
Mitch Marner also scored for Toronto, while Joseph Woll made 25 stops.
The Flames got tagged for too may men just 1:13 into the game. Mikael Backlund picked up the puck in the Flames zone and then drove around a defender and across the Toronto crease, Woll stopping his shot and then making a stellar blocker save on his rebound attempt.
But the Flames kept coming, Coleman then getting a shorthanded breakaway after some great PK work down low and making no mistake with a patient shot high farside at 2:32. It was the Texas product's 14th of the season and 101st career tally.
Before the powerplay ended, Markstrom was quick across his crease to stop a cross-ice one-timer from Mitch Marner, who was parked in the left faceoff dot.
With just under seven minutes to go in the frame, Noah Hanifin took a pass and walked in to the high slot and rifled a wrister that went off the near post, some fans rising from their seats thinking it was a goal.
Coleman nearly got his second early in the middle stanza. Rasmus Andersson's shot missed the net high but bounced off the glass and back out front above the crossbar, Coleman showing some great hand-eye coordination to tip it out of the air at Woll's feet but he made the stop.
Marner tied it up at 7:36 of the second putting a low shot past Markstrom.
Kadri appeared to put the homeside ahead again with a long-distance wrister that beat Woll gloveside on the powerplay just over a minute later with Marner in the box for highsticking Chris Tanev, but the Leafs challenged saying the play was offside and after review, the officials agreed and overturned the tally.
Markstrom was called upon not long after on a Leafs shorthanded rush, getting a piece of newcomer Jake McCabe's offering with his glove to deny him and then Alex Kerfoot lofting the rebound over the cage.
MacKenzie Weegar then got a lovely look in tight but Woll flashed the leather to deny him.
Then it was Toronto's turn for some iron, Marner ringing one off the far post, followed by Markstrom again coming across to deny a one-time blast from Auston Matthews.
But the Flames 'tender saved his biggest stop of the period for the final minute of the second, stoning William Nylander's breakaway attempt to keep it knotted at one after 40 minutes.
Markstrom stopped Ryan O'Reilly on a breakaway in the third, but Jarnkrok was there to jam the rebound home at 2:20 for the game-winner.
