It was the fourth shutout of the season for Varlamov, who has 17 in his NHL career.
The Avalanche (21-18-11) have won four consecutive home games, are 12-5-5 overall since Dec. 11 and have closed within two points (55-53) of the Calgary Flames for the second wild-card spot in the Western Conference. The Flames have played one fewer game and have a 23-15 lead in regulation and overtime wins.
The Predators (31-11-6) are 0-1-1 in the first two games of a three-game road trip and 2-2-2 since goalie Pekka Rinne sustained a knee injury Jan. 13 against the Vancouver Canucks. Marek Mazanec made 25 saves in his first start and second appearance of the season.
"That was our most complete game of the season," Avalanche coach Patrick Roy said. "We played 60 minutes. From behind the bench it was fun to watch."
Nathan MacKinnon, who had two assists, set up Gabriel Landeskog for an empty-net goal with 1:11 remaining in regulation to seal the win. The Avalanche also got a goal each from Jarome Iginla, who tied Dino Ciccarelli for 46th place on the NHL's all-time list with his 1,200th career point, and Maxime Talbot.
Varlamov has gone 8-3-2 with three shutouts since he returned Dec. 23 from his third groin injury of the season. He's allowed two goals or fewer in eight of those games.
"I felt pretty good today," said Varlamov, who made 35 saves Tuesday in the Avalanche's 4-3 overtime loss in Nashville. "We had two days between the games, and that helped a lot. I wasn't happy with my game three nights ago, a couple of bad goals I gave up. I had a meeting with my goalie coach (Francois Allaire) and we were talking about my game, and he reminded me of a couple of good things."
Said Roy: " It's nice to give him a performance like this. I'm not saying it was an easy shutout, but it's nice for him to have an easier game where we don't give that many shots and not many scoring chances. I hope we're going to do that all the time from now on."
Nashville coach Peter Laviolette credited Varlamov for playing well, but he said the Predators didn't make it especially difficult for the goalie.
"He made some saves, but we could have done a better job getting traffic in front of him," he said.
Iginla put the Avalanche in front 2-0 at 5:54 of the third period, six seconds after the Predators killed off Mattias Ekholm's penalty for hooking MacKinnon. The Avalanche went 0-for-4 on power plays, stretching their slump to 0-for-17 in the past four games.
Iginla was in the slot when he accepted a pass from Matt Duchene and took a quick shot past Mazanec with Predators defenseman Seth Jones in front. Iginla has scored a goal in three consecutive games and leads the Avalanche with 16.
"It's cool," Iginla said of his milestone point. "Honestly, I definitely feel fortunate. When I started, I wouldn't have guessed that. You know, I just wanted to get in the League and I was so excited to be in the NHL, and then you just want to stay in the League for a while. I've been so fortunate, blessed, to play a long time with some great players and be on power plays, play in some great cities and stuff. It's been fun."
Iginla 35 goals and 25 assists against Nashville in 55 games.
"It was a really good game for our team," he said. "We moved the puck well, we were strong defensively and we were assertive and we are building on some stuff we're trying to do. It's a big game; it's about momentum and trying to keep that coming."
The Predators went on their second power play of the game at 18:41 of the second period when Shea Weber was hooked by Avalanche forward Cody McLeod. The advantage carried into the third period, and Colorado finished killing it off.
McLeod became the fourth player in Quebec Nordiques/Avalanche franchise history to reach 100 penalty minutes in a season seven times.
Center Mike Ribeiro, the Predators' leading scorer, was given a minor penalty for unsportsmanlike conduct, a 10-minute misconduct and a game misconduct at 6:33 of the third period.
One of the Predators' best chances came at 6:15 of the second when Gabriel Bourque skated in alone, but Varlamov was able to poke the puck away with his stick before Bourque could take a shot.
"I had the breakaway and tried to shoot it with my backhand," Bourque said. "It went right to him. It was a night where their goalie had a good night. We had some good minutes where we played well. We just didn't get the results."
The Avalanche took a 1-0 lead at 13:49 of the first period on Talbot's goal. Ryan O'Reilly passed from the left side to MacKinnon near the right crease for a shot that Mazanec stopped. The rebound bounced into the slot, and Talbot swept it into an open net.
The Predators, who lost 5-4 in a shootout Thursday to the St. Louis Blues, are 1-3-2 in the second game of back-to-back sets.
"It wasn't our best game," Nashville center Matt Cullen said. "We just have to be better."