Jan Hejda, Gabriel Landeskog and Matt Duchene each scored a goal for Colorado, which is 8-1-1 over its past 10 games. The Avalanche earned their first win in Nashville since Jan. 6, 2009, ending a 0-6-1 skid.
Colorado center Paul Stastny said they did a good job surviving late, with the key jumping out to a 3-0 lead in the first.
"Getting it to 3-0 was important for us especially in the past not doing well here," Stastny said. "Knowing we have a new team, new coach and new mentality I think we forgot about playing here and a fresh start for us."
Varlamov made 27 saves and improved to 11-0-5 in his past 16 games, tying the Avalanche record for consecutive games without a regulation loss.
The Predators scored three straight goals after trailing 5-1 in the third, and Roman Josi's goal at 17:03 gave them a chance to force overtime. But Varlamov stopped a flurry in the final minutes, including shots from Josi and Mike Fisher for the win.
"It's not important how the game ends," Colorado coach Patrick Roy said. "It's the result at the end. We won the game."
Patric Hornqvist, Craig Smith and Ryan Ellis also scored for Nashville, which snapped a two-game winning streak. Devan Dubnyk struggled in his Nashville debut after the Predators swapped forward Matt Hendricks to Edmonton for the goalie Wednesday.
"He got nervous and was rusty," Nashville coach Barry Trotz said. "I know Dubnyk has been in the league a long time, and I've seen a lot better games from him. But it was his first game. We gave him very little support early, and it just looked like he was nervous after that."
Colorado took control from the start with three goals in the first 12:05 with Holden beating Dubnyk with a wrister from the left circle off a rebound at 6:21 in the first.
Then the Avalanche scored twice 38 seconds apart with Hejda beating Dubnyk with a slap shot through traffic at 11:27, then Landeskog scored on a one-timer from the right circle at 12:05.
Trotz then took his timeout for Nashville.
The Predators got on the board in the second after Landeskog elbowed Ellis high, drawing a penalty. Erik Johnson drew a roughing penalty when he fought with Paul Gaustad, giving Nashville a 5-on-3 advantage for 2 minutes. Hornqvist scored on a tip-in 35 seconds later in pulling Nashville within 3-1, but the Avalanche killed off the rest of the power play.
The Predators thought they beat the buzzer with a slap shot from captain Shea Weber only to see officials waive it off because the clock expired before the puck crossed the line.
Holden scored his fourth goal of the season and second of the game on a slap shot at 2:31 of the third with the puck going in off Dubnyk's left skate. Holden then had the primary assist on Duchene's power-play goal at 6:31, which wound up as the game winner.
The defenseman credited his Colorado teammates for his big night.
"I think every game someone else is stepping up," Holden said. "We've gone through a lot of injuries. It's showed the adversity of our team here to battle through it. Every night is a different guy, and tonight it was luckily me."
The Predators tried to rally as Smith added a goal at 8:14 tapping in the puck behind Varlamov. Then Ellis scored on a slap shot from the point at 12:45, and Josi's goal gave Nashville a chance. But even with Dubnyk pulled, they couldn't beat Varlamov again.
"We just didn't win enough battles in the beginning," Josi said.
Peter Budaj went 13-0-3 in a 16-game span between Feb. 24 and April 5, 2007, for Colorado. ... Stastny, Nathan MacKinnon and John Mitchell each had two assists. ... Stastny had missed the past two games with a leg injury. ... The Predators have allowed at least three goals in 28 games this season. ... Nashville has five goals on the man advantage in the past three games and came into the game sixth in the NHL on the power play. ... Smith's goal gave him a career-high 15 this season.