Varlamov improved to an NHL-leading 6-0 on shootouts this season. Colorado, which erased a 1-0 lead with fewer than six minutes to go, has won 10 straight shootouts dating to last season.
"Nothing special," Varlamov said. "I don't know, maybe good luck. I don't like shootouts. We play shootouts a lot. I just try to stay on the ice and wait to see what the player going to do. Just wait, wait, wait.
"Thank God we win tonight."
The Avalanche have won eight of their past nine games, including shootout wins against Los Angeles and Philadelphia.
"I think goaltending's the main thing," Ryan O'Reilly said. "After a couple of wins there, you start to sense confident there on the bench. Guys know we’re going to get the extra point. You got Hedjuk shooting. I don't think he misses too much. Varm in the pipes stops them every time."
Varlamov stopped 31 of 32 shots in overtime and regulation and outdueled Quick, who was named the NHL's third star of the week.
Quick was six minutes away from his sixth shutout this season when O'Reilly snapped in the puck from the right side to tie it at 1 after the Kings' fourth line couldn't get the puck out of its zone.
After Stoll missed the net on L.A.'s first attempt of the shootout, Varlamov stopped Anze Kopitar and Dustin Brown. Hedjuk slipped the puck through five-hole on Quick.
Colorado shooters are 12 for 19 in the shootout this season.
"It's definitely a character win for us," O'Reilly said. "They're a great team. To win in this situation is a huge boost for us."
Los Angeles has gained points in seven straight under coach Darryl Sutter (4-0-3) and has still not suffered a regulation loss in his tenure.
The Kings have not allowed a power play goal in 21 attempts under his watch and have killed 27 straight penalties. That streak was kept alive thanks to Quick, who robbed TJ Galiardi during a third period Colorado power play that seemed to be Colorado's last gasp before O'Reilly's score-tying goal.
But the Kings can lament a squandered power play with 1:35 remaining in overtime. They also missed a power-play opportunity with 2:32 to go in regulation when Stefan Elliott hooked Drew Doughty, but Doughty was called for diving on the play.
Sutter praised his team for playing hard on their fifth game in eight nights. He also saw some inconsistency.
"We just stopped shooting the puck in the second," Sutter said. "We started to make that little play across the net, those sort of things in the second period, instead of putting it to the net and getting to the net. I like the percentages when the goalie has to make the save."
Colorado scored four goals in a victory against Anaheim on New Year's Eve, but looked out of sync for most of the night in its second game without second-leading scorer Matt Duchene, who is out for about a month with a knee injury.
On its only power play of the game, Colorado made an errant back pass and Galiardi made the blooper reel when he tripped with the puck at center ice.
L.A.'s new top line, on the other hand, produced on its first shift of the second period.
Mike Richards cycled the puck to Kopitar behind the goal line and the latter's rebound found Brown, who swiped a backhand through Varlamov's pads for a 1-0 lead.
Both sides had quality chances in the second but Quick and Varlamov held ground, including a terrific right pad save by Varlamov on Jack Johnson during a Kings' power play.
It was scoreless after the first period, but the Kings continued to start games well under Sutter. They enjoyed a 12-5 shot advantage and were in Colorado's zone for so long at one point they were able change out Andrei Loktionov's line for Jarret Stoll's.
"I disagree with the way we handle that," he said. "I think every puck that goes in the net, unless the whistle went, should be called a goal and because of what we have in Toronto, then they should rule from there. We talk about goal scoring and offense in the League and all those things. Every puck that goes in the net should be called a goal, and then they can look at it. To me, it makes more sense that way."