Varlamov signed a five-year, $29.5 million contract extension before stopping 31 shots in a 5-4 win against the Minnesota Wild at Pepsi Center.
Varlamov established a career high with his 27th win and set a franchise record with his 10th win in January as the Avalanche solidified their third-place standing in the Central Division.
"I'm excited," he said. "It's great news for me. I'm happy. I want to say thank you to the organization to give me a chance to play for this team. Thanks for my teammates. I think they have played very well for me. That's why I signed that contract, because they helped me."
Colorado (34-14-5) finished the season series with a 4-0-1 record against fourth-place Minnesota (29-21-6), and increased its lead to nine points (73-64) with three games in hand.
"It was a big game for us," said center Matt Duchene, who had two assists. "We have three games in hand, and we're up [nine] points in the standings. That puts us well ahead, and we just have to keep taking care of business."
"Our fans want entertaining games, that's what it is," Avalanche coach Patrick Roy said. "We made it pretty interesting for them. It's just the small details like make sure we keep our focus. You're not always going to win the 5- or 6-1 game when you're up 4-1, especially if you lose your focus or take it a little bit easy."
The Wild closed within 4-3 in the third period on goals by Zach Parise, who had two goals and two assists, and Mikael Granlund. Colorado's Nathan MacKinnon scored on a breakaway to give the Avalanche a 5-3 lead with 2:25 to go, but Minnesota's Jason Pominville answered 11 seconds later.
"They had a couple guys that skated really well and gave us a tough time, but it was what we weren't doing," Pominville said. "We weren't skating enough, we weren't on our toes enough early in the game. I mean, when you get behind the 8-ball like we did, it's tough to get back. They are a good team; they are a good team for a reason. They do a lot of good things, and we got away from what we do best."
Third periods have been a problem for the Avalanche in recent weeks. After being dominated in the second period, the Wild outshot the Avalanche 17-4.
"Tough game," Varlamov said. "I wish we could win 4-1 or 5-1. We always lead 2-1, 3-1, 4-1, and then give up goals in the third period. Maybe I have to be better than this, stay more focused and stay sharp in the third period."
Parise beat Varlamov with a shot past his glove at 7:47, and Granlund scored at 12:51. After Varlamov made a save against Keith Ballard, Parise got to the rebound and passed across to Granlund, who scored into a wide-open net.
"The best part about it was we started to play the right way and were awarded for it," Wild coach Mike Yeo said. "We definitely looked like a different team. We had much better focus going into the third period."
Mitchell scored at 1:43 of the second to break a 1-1 tie shortly after Jamie McGinn put a big hit on Wild defenseman Nate Prosser in front of the Avalanche bench. Nate Guenin passed to Mitchell, who skated to the top of the faceoff circles and shot the puck between goalie Darcy Kuemper's pads.
Stastny put in the rebound of Erik Johnson's shot at 9:17. Stastny passed back to Johnson and moved to the front of the net. Johnson fired a shot from above the right circle, and Stastny put in the loose puck.
Stastny sustained what appeared to be an ankle injury in the third period. He will be re-evaluated Friday.
Talbot converted another rebound at 15:26. MacKinnon slid a cross-ice pass to Mitchell, who one-timed a shot from the right circle. Kuemper deflected the puck with his glove, the puck dropped outside the crease, and Talbot was credited with the goal while battling with Minnesota's Kyle Brodziak in front.
"It was frustrating," Brodziak said. "The rebound came out, and I got it on my stick and was real slow reacting. The guy was right on my back. My leg went forward and I ended up putting it in."
The Wild replaced Kuemper with Niklas Backstrom after that goal. Kuemper faced 23 shots, 13 in the second period.
"I let him know it wasn't him," Yeo said of Kuemper. "I did it to try to change the momentum at that point."
Ryan O'Reilly gave the Avalanche a 1-0 lead at 4:41 of the first period with his 20th goal, two more than his previous career high set in 81 games in 2011-12. Matt Duchene passed to O'Reilly in the right circle for a shot that beat Kuemper to the far corner.
O'Reilly has six goals and three assists in the past nine games after missing two games with a shoulder injury.
The Wild tied the game at 6:16 on a power-play goal by Parise. Minnesota had gone 1-for-26 with the man advantage in the previous nine games, but scored 26 seconds after Gabriel Landeskog was penalized for hooking.
Parise has three goals and four assists in four games since returning from a foot injury that caused him to miss 14 games.