Jake Muzzin got his first goal of the season and Auston Matthews also scored for Toronto. Mitch Marner added two assists and Jack Campbell stopped 31 shots for his second win in two starts this season.
Two Toronto goals caromed off Flames bodies and another off a Leaf skate.
''It was a pretty ugly game to be honest for us,'' Matthews said. ''I think there was a lot of things we could have done better.
''Obviously a couple of solid, lucky bounces that went our way.''
Down 3-2, Calgary pulled Markstrom for an extra attacker with 1:49 remaining. The Flames also called a timeout with 37 seconds to play but couldn't produce the equalizer.
''That's hockey. Bounces go each way throughout a game,'' Monahan said. ''We've got to clean up a few areas and I think we've got to be around their net a little bit more to get those bounces.''
Lindholm pulled the Flames within a goal on the power play at 6:33 of the third period. With Matthew Tkachuk providing a screen in front of Campbell, Lindholm's wrist shot from the top of the faceoff circle beat Toronto's goalie short side.
Matthews returned to the Maple Leafs' lineup after sitting out Friday's 4-2 win over the visiting Edmonton Oilers with upper-body soreness.
''I'll obviously have to take care of it in the next couple of weeks, but it feels a lot better and feels more than good enough to play and good enough to go out there and contribute,'' Matthews said.
He was in the slot waiting for a play to develop when the puck deflected off him and Flames defenseman Rasmus Andersson for a goal at 2:42 of the third period.
Simmonds, who signed with Toronto as a free agent in October, earned his 500th career NHL point and gave the Leafs a 2-1 lead late in the second period. Simmonds was parked in front of Markstrom when what appeared to be his between-the-legs backhand pass deflected off the inside of his right skate and past Markstrom for a power-play goal 32 seconds before the second-period buzzer.
The Maple Leafs remain in Calgary for Tuesday's rematch before heading to Edmonton on their four-game road trip.
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