Anderson matched a season high with 48 saves, Paul Stastny scored twice, and the Avalanche rallied to beat Dallas 5-3 on Sunday.
"Our goaltending was outstanding this afternoon, just like it's been all season," Colorado coach Joe Sacco said.
The Stars turned up the pressure in the final minute with a 6-on-4 advantage when they had a power play and pulled their goaltender, but Anderson responded with a series of sprawling saves to help his team take the season series from Dallas 3-1.
Anderson also got help in the final seconds from defenseman Kyle Cumiskey, who swept a rebound away from the net while Anderson was prone on the ice.
"I was just trying to make myself big and follow the puck off my body," Anderson said. "Sometimes you get a little lucky. Our D was doing a good job protecting me."
Brandon Segal scored twice in a span of 1 minute and 22 seconds of the second period for his first career multigoal game, giving the Stars a 3-2 lead.
Stastny and Chris Stewart answered with goals during a stretch of 2 minutes and 41 seconds of the second to put Colorado in front.
Stastny and Tucker scored second-period power-play goals. Colorado is 8-for-20 with the man advantage over the past six games.
Colorado is 21-11-4 when it scores at least one power-play goal. Stastny said his team's recent success with the man advantage comes down to a basic approach.
"I think we're shooting and getting more shots through," Stastny said. "You can't get too cute. Sometimes when you get shots off, that's when plays open up."
Stastny's goal from beyond the right circle at 12:23 of the third period pushed the Avs' lead to 5-3.
Milan Hejduk added a goal for the Avalanche, 4-3-0 since the Olympic break to maintain their runner-up spot behind Vancouver in the Northwest Division.
"It's really frustrating," Segal said. "It's definitely bittersweet to get two goals and lose. We need wins right now, that's the most important thing. It's a do-or-die situation for us right now."
Anderson was the difference, withstanding a 37-shot barrage in the final two periods to improve his record to 35-19-5.
"Anderson's been one of their MVPs," Stars captain Brenden Morrow said. "He gives them a chance to win every night."
Hejduk's wrist shot from the right circle got through Dallas goaltender Marty Turco's pads to give Colorado a 1-0 lead at 10:19 of the opening period.
Benn drew the Stars even at 1 at 14:31 of the first period, shoveling a shot into an open net off a centering pass from James Neal.
Tucker put Colorado in front 2-1 at 4:28 of the second period on a one-timer from the slot with 34 seconds left on a tripping penalty to Dallas' Trevor Daley.
Segal's goal 1 minute and 31 seconds later tied it, and he gave Dallas the lead at 7:21 on a one-timer from the top of the right circle.
Stastny's power-play deflection at 11:05 of the second period made it a 3-all game, and Stewart's wraparound at 13:46 gave the Avalanche a 4-3 edge despite Dallas' 20-12 shots advantage in the period.
At that point, Stars coach Marc Crawford replaced Turco with Kari Lehtonen, playing his third game since being acquired in a trade with Atlanta on Feb. 9.
"I thought that was the call that needed to be made," Crawford said of the goalie change.
The Stars were playing the second game of a six-game homestand. ... Stewart has four goals in four games against Dallas this season. ... Stars D Nicklas Grossman (back) missed his fourth straight game. ... Colorado C Stephane Yelle and LW Cody McCloud sustained leg injuries in the first period and did not return. Sacco said he'll know more about the extent of the injuries on Monday. ... The Avs are also without D Ruslan Salei (torso), RW Brandon Yip (shoulder) and RW David Jones (knee). ... Segal, claimed off waivers from the Los Angeles Kings on Feb. 11, has four goals and two assists in his past six games. ... The game wasn't televised in Dallas-Fort Worth, breaking a streak of 1,101 Stars games (including playoffs) that had been available on local TV. It was originally scheduled as an NBC "flex" game, but the network decided to show Washington-Chicago instead and no local station picked up the game.