Varlamov stopped 34 shots, Barrie scored 50 seconds into overtime and the Avalanche beat the Nashville Predators 1-0 on Saturday.
Pekka Rinne had 23 saves for Nashville, which has lost two straight but gained a point heading to Chicago on Monday.
"Any point on the road is a good point," coach Barry Trotz said. "They're playing pretty well, they've got a lot of offense and they're at home. Obviously you want two points but we'll take one."
The Predators are fighting to make the playoffs with a month left in the season. They came into Saturday two points behind St. Louis for the eighth and final postseason spot in the Western Conference with 14 games remaining. They got one point despite being shut out for the sixth time this season and wasting a strong outing by Rinne.
"We had chances, and they were good chances," defenseman Shea Weber said. "We've got to score goals if we want to win."
They failed to do anything against Varlamov, who recorded his third shutout of the season and No. 11 for his career. He came up big throughout the game to give Colorado a chance to win in overtime.
"Varly has been unbelievable," Avalanche center Matt Duchene said. "He's been so consistent for us. It was one of those games that felt like the first goal would win it."
They got the one goal early in the extra period. Barrie scored when he came down the right side and sent Greg Zanon's rebound past Rinne. It was Barrie's second goal of the season and ended a four-game skid for the Avalanche.
"I thought Z might one-time that, and if Rinne made the save it might pop out to me," Barrie said. "I was fortunate to get the bounce and it was pretty much an empty net. I made it a little harder than it was and put it in off his pad."
It was just the third goal by a Colorado defenseman this season, two of them from Barrie.
"Obviously we've been struggling to score goals from the back end, myself included, but it was definitely nice to get that one," Barrie said. "Any time we can get an OT winner it feels pretty good."
The Avalanche have had few reasons to celebrate of late. Colorado was 10-10-4 after beating San Jose on March 10, but has gone 2-8-0 since.
The Avalanche were in the hunt for a playoff berth three weeks ago, but are now on track for a high draft pick for the fourth time in the past five years.
"It's been a tough stretch here this last little while, but not too many times did we talk about or have complaints about effort or work," Colorado coach Joe Sacco said. "This afternoon we got rewarded."
Nashville outshot Colorado 27-12 in the first 40 minutes, in part due to six power-play opportunities, one of which was a double minor for high sticking.
Nashville wasn't able to take advantage even with a two-man advantage for 33 seconds. Varlamov was sharp and kept the game scoreless.
"We had a lot of chances on the power play and we weren't able to convert," Trotz said. "It was Varlamov. Most nights one of those will go in."
The Avalanche had just one power-play opportunity, which lasted 14 seconds. ... Nashville center Mike Fisher was scratched due to an upper-body injury suffered in Thursday's loss to Phoenix. He is considered day to day. ... Avalanche C Paul Stastny (foot) missed his third straight game. He is expected to be out a total of 10 days. ... Colorado forward David Jones was scratched. He's day to day with a knee injury. It was his third game missed with the injury and fourth straight game he has not dressed.
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DENVER, CO - MARCH 30: Gabriel Landeskog #92 of the Colorado Avalanche warms up prior to facing the Nashville...