The shutout was Anderson's fifth this season and his second in three starts. He has allowed just two goals in the past four games, stopping 117 of 119 shots.
"We've just been playing great the last few games, making my job a lot easier," Anderson told reporters.
"We're making plays in the offensive zone and getting a lead," he said of the Avalanche's longest winning streak since March 2008.
Brett Clark's powerplay score with the game five minutes old and T.J. Galiardi's shorthanded tally six minutes later staked the Avalanche to a 2-0 first period lead.
Stewart made it 3-0 with his 17th goal of the season in the second and Stastny added the final score off a powerplay in the third.
"I looked up and thought it was an icing," Stewart said of his team-leading goal. "Then it (the puck) hit the ref, that put us on a 3-on-2 and (Wojtek) Wolski to Stastny and Stastny to me. I just tried to put it on net and it beat (Stars netminder Marty) Turco."
Stastny said he was also fortunate to score.
"I saw the shot got tipped and I was trying to throw it in front of the net, and I just got lucky," he said.
Dallas, the last-placed team in the Pacific Division, has won only once in its past 12 road games.
"We need an attitude adjustment," said Turco, who had 32 saves. "We have to trust our coaches to help us figure it out."
(Reporting by Gene Cherry in Salvo, North Carolina. Editing by Nick Mulvenney)