The Flames overcame a 47-save performance by Budaj and earned a 4-1 win over the Colorado Avalanche on Tuesday night.
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.
"It was big on so many levels just because of the way we've played our last few games," said Aucoin, who scored the go-ahead goal in the third period. "Goals-against have been killing us. Penalties have been killing us.
"We stayed out of the box and played good defense and we put a lot of pucks on net with a lot of traffic, which is something we haven't been doing enough of. It was exactly what we needed."
It looked as though Budaj would steal the game until Aucoin fired a slap shot over the goalie's right glove at 12:28 to put Calgary up 2-1.
Iginla scored an empty-netter, and Moss added a power-play goal with under 10 seconds remaining. Miikka Kiprusoff made 22 saves for the victory.
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 made several pad saves from close range.
"I think their coach said to throw everything at the net," Budaj said. "They're a big, physical team and they put the puck in the corner and crash the net.
"We didn't win the battle along the walls and that's why they had a lot of shots."
Aucoin said he and his teammates knew they had to keep the pressure on Budaj.
"Our game plan was pretty much knowing he's won some games for them that way, and we knew he was going to make big saves," Aucoin said. "It's not the old Colorado where they have tons of offense. They play a hard defensive game, and we just had to make our own opportunities. We knew it wasn't going to be right away.
"Sooner or later it was going to happen because our forwards did such a great job."
The Avalanche were without captain Joe Sakic (back) for a fourth straight game, but the team hopes he will return Thursday at home against the Flames.
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 served only two minor penalties. They were the more rested team with four days off following the San Jose blowout. Calgary had spent that time shoring up its defensive coverage.
"You could tell guys were fresher, our system was a little smarter, and mentally we were rested, which made a big difference," Aucoin said. "We played the way we wanted to, and so many nights you want to play the right way. It's not that easy, but tonight we made it look a little easier."
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 6:52 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 Kiprusoff over his right shoulder for his third goal.
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.