Marek Svatos had the lone goal for the Avalanche, whose winning streak ended at three. Colorado will head home after going 2-1 on a road trip that included shootout victories at Vancouver and Edmonton.
Budaj shouldered the workload for Colorado, making 47 saves. Calgary peppered him with a flurry of shots, many of them from close range.
Calgary, which improved to 4-5 in November, had lost five of seven before Tuesday - including an embarrassing 6-1 defeat in San Jose last week.
The Avalanche were without captain Joe Sakic (back) for a third straight game, but the team hopes he will return Thursday at home against Calgary.
Aucoin took two steps in from the blue line and fired the puck over Budaj's glove for the go-ahead goal at 12:28 of the third period.
Calgary's power play was 0-for-7 until Moss scored in the final six seconds of the game while Ryan Smyth was in the penalty box for unsportsmanlike conduct.
The Flames opened the third period with a two-man advantage, but couldn't muster a goal. Calgary outshot the Avs 32-15 after two periods, but the game was deadlocked because of Budaj.
Calgary tied it at 6:53 of the second period. A sprawling Mike Cammalleri managed to bank the puck across the goal-mouth to Bourque, who scored high with a sharp-angled shot.
Colorado took a 1-0 lead at 6:48 of the first period. T.J. Hensick backhanded the puck from behind the goal line out to an unchecked Svatos, who beat Miikka Kiprusoff over his right shoulder for his third goal.
Notes: Calgary is 2-0 against the Avalanche this season. ... In addition to Sakic, Colorado was also without forward Ben Guite (ribs). ... Forward Matthew Lombardi (shoulder) and defenseman Jim Vandermeer (ankle) were out of the Flames' lineup.