Matthew Tkachuk ended an eight-game goal drought, Rasmus Andersson had two assists, and Jacob Markstrom made 24 saves for the Flames (15-13-3).
Calgary was coming off its only loss since Sutter made his debut March 11, 7-3 to the Edmonton Oilers on Wednesday. He replaced Geoff Ward, who was fired March 4.
Jason Spezza, Alex Kerfoot and Mitchell Marner scored, and Frederik Andersen made 14 saves for the Maple Leafs (19-10-2).
"We got a pretty good group in there," said Toronto forward Wayne Simmonds, who returned after missing 18 games with a broken wrist. "We know we're doing some good things, but at the same time, we're having these little mistakes that are ending up in the back of our net. We've got to make sure that we're playing a full 60 minutes; not 50, not 55."
Andersen is 1-5-0 with a 3.84 goals-against average and .862 save percentage in his past six games.
"It's difficult on a night like tonight when everything seems to find its way through traffic," Andersen said. "I know personally I didn't lose my abilities to play good. Same goes for our team. You want to get out of this slump here and start winning games."
Mark Giordano gave the Flames a 4-3 lead at 12:33 of the second period, three seconds into a power play, when his shot from the point went in off defenseman Jake Muzzin in the slot.
Calgary was 3-for-3 on the penalty kill, including two in the third period after Mikael Backlund was called for slashing Simmonds at 7:33 and Derek Ryan for holding Zach Hyman at 12:06.
"Definitely the penalty kill (felt better than the power-play goal) for me," Giordano said. "They both feel really good for me, not going to lie, I've got to be honest. But it's a pretty good power play over there, they've got some elite players who can make plays, and I thought we did a good job of limiting them and trying to block shots when we could and get in lanes when we could."