Trailing 2-0 after the first period, the Flames tied the game 2-2 on a goal by Elias Lindholm with 1:54 left in the third period. Lindholm scored his fourth goal with a shot from the left circle shortly after Calgary goalie David Rittich went to the bench for an extra skater.
Colorado's Nathan MacKinnon matched the franchise record for the longest season-opening goal streak, five games, when he scored 11 seconds into the first period for a 1-0 lead. The record was set in 1992-93 by Mats Sundin of the Quebec Nordiques.
MacKinnon drove to the net to put in the rebound of Mikko Rantanen's backhand shot. Rantanen has at least one assist in each of the first five games, and has seven this season.
Sam Bennett also scored for the Flames (3-2-0), who went 2-1-0 on a three-game road trip. Rittich made 24 saves.
"We knew we had to be better," Gaudreau said of the Flames' poor start. "We came in here after the first, talked about it, said there's lots of time left, and we showed what we're capable of."
The Flames outshot the Avalanche 31-9 after the first period, 41-26 for the game.
"You can't just dominate the first period and then the rest of the game … you have to play all 60 minutes," Varlamov said. "In these kind of games, we should be happy, probably, with the one point because of the way we played."