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.
Joe Sacco's decision to shake up his lineup paid immediate dividends for theColorado Avalanche coach.
It was a banner return to action for Olver, who had been a healthy scratch for seven games. He used the opportunity to state his case for more playing time.
Matt Duchene pounced on a loose puck just in time to turn it into a game-winning goal.
Duchene corralled a deflection from teammate P.A. Parenteau's desperation, long-distance shot and scored his second goal of the night as time was about to expire in overtime, lifting the Colorado Avalanche over the San Jose Sharks 3-2 on Sunday to stretch their home winning streak to a season-high five games.
"It feels awesome. I didn't think it was going to count," Duchene said. "Obviously, you're really excited at first but you're holding back a bit in case it doesn't count."
Officials upheld the goal after a replay review showed Duchene got the shot past goalieAntti Niemi an instant before time would have run out in the extra period.