Valeri Nichushkin set up MacKinnon with a feed at the Calgary blue line to cap off a three-goal comeback in the final 9:40 of the third.
Mikko Rantanen had a goal and two assists, and MacKinnon, Cale Makar and Ross Colton each had a goal and an assist for the Avalanche (17-9-2), who lost five of their previous six games (1-3-2).
Alexandar Georgiev allowed five goals on 22 shots before he was replaced by Ivan Prosvetov at the start of the third. Prosvetov made 12 saves.
Nazem Kadri and Blake Coleman each had a goal and an assist for the Flames (11-14-3), who have lost four of five (1-4-0). Dan Vladar made 31 saves.
Colton cut it to 5-4 at 11:20 of the third after Josh Manson’s point shot caromed off the skates of Sharangovich and Zary and landed on his stick.
Rantanen tied it 5-5 at 13:31 when he deflected Makar’s shot from the right circle.
Kadri gave Calgary a 1-0 lead at 13:46 of the first period with a wrist shot from beyond the left circle after a giveaway by Rantanen in the neutral zone.
Tomar Tatar tied it 1-1 at 18:14 with his first goal of the season after he sent Colton’s pass from behind the net over Vladar’s blocker.
Makar gave Colorado a 2-1 lead on a power play at 1:43 of the second period, one-timing a centering pass from Rantanen to the far side.
Andrew Mangiapane tied it 2-2 at 5:14 with a one-timer from between the circles that beat Gerogiev’s blocker.
Coleman beat Georgiev with a snap shot from above the left circle at 13:21 to give the Flames a 3-2 lead.
Ben Meyers tied it 3-3 at 14:57 when he took a pass from Andrew Cogliano inside the right circle and scored with a wrist shot to the far side.
Meyers was recalled from the Colorado Eagles of the American Hockey League earlier Monday.
Connor Zary put Calgary back in front 4-3 when he flipped a rebound up in the air and then batted it in at 16:19.
Yegor Sharangovich tapped in a cross-crease pass at 18:34 to make it 5-3 after Noah Hanifin circled the net with the puck.
Flames defenseman Chris Tanev left 15 seconds into the game after a hit from Colton. He did not return.
