Milan Hejduk and Chris Durno, with his first NHL goal, also scored for the Avalanche. Alex Auld made 29 saves, and Steve Ott scored for the Stars, who suffered just their third regulation loss in December.
Budaj played an average of 45 games the previous four seasons but has become the forgotten backup with the stellar play of No. 1 goalie Craig Anderson. Budaj made just his seventh start of the season against the Stars, but he made it memorable with one of his best games in his five-year career.
"He played excellent," Colorado coach Joe Sacco said. "He made some keys saves down the stretch."
Budaj kept Dallas off the boards for most of the game with two big stops on defenseman Jeff Woywitka. The first came on a rebound late in the second and he made a glove save of Woywitka's shot on a Stars' 4-on-2 rush with 16:25 left in the third.
"There was an unfortunate turnover for us at the blueline and they made a nice pass to Woywitka and he made a good shot," Budaj said. "I tried to be as big as possible and I was able to react."
While Budaj was turning away the Stars, his teammates were building a lead.
Durno gave Colorado the lead 6:04 into the game. Durno took a pass from Matt Hendricks behind the Stars' net and he banked a shot off Auld's stick into the net for his first NHL goal.
"It felt amazing," Durno said. "It's something that you work to get for a long time."
Colorado continued to pressure Dallas throughout the first period, putting 14 shots on Auld and creating several good scoring chances. The Stars' best chance to score in the first came late when James Neal's shot wide of the open net.
"I just shot it over the net," Neal said. "I'd like to have it back but you can't do anything about it now."
Mike Modano also had a chance to score when Budaj was out of position but he whiffed on a shot.
"It looked like the hockey gods were shining on them tonight," Auld said.
The Avalanche didn't have a shot on goal for the first 14 minutes of the second period and then cashed in on their second one of the frame. Matt Duchene sent a pass from behind the net to Hejduk, who was skating down the slot. The winger buried the puck for his 15th goal of the season with 5:56 left in the second.
"We did some selfish things tonight," Stars coach Marc Crawford said. "We were too long on shifts, and that resulted in the second goal. We get trapped in, pinching late in the game instead of staying in. When we put ourselves behind and we get behind, you make the game more difficult than it has to be."
Dallas finally broke through with 12:16 left when Ott scored on a backhander, his fifth goal of the season. The sellout crowd grew anxious at the memory of Colorado blowing a 2-0 third period lead to Anaheim on Tuesday night, but the fans breathed easier when Stewart scored his 10th goal on a 2-on-1 rush with 6:26 left, and he added an empty-net goal with 58 seconds remaining.
"It's been our tendency of ours to give up couple goal leads, so it was good to go up 3-1 and then 4-1," Stewart said.
The Stars pulled Auld with 2:03 left, but managed just one shot on goal with the extra skater.
NOTES: Avalanche video coordinator Adam Deadmarsh was honored before the game as part of the organization's yearlong salute to past stars. ... Colorado is 0 for 11 on the power play in its last three games. ... The Stars lead the NHL with 11 shootout/overtime losses.