Varlamov made 19 saves in winning his third straight start, and Jamie McGinn provided all the offense 2:32 into overtime to lift the Avalanche to a 1-0 victory on Saturday night.
Varlamov's 12th career shutout and first this season came one day after prosecutors in Denver filed a misdemeanor assault charge against him following a complaint by his girlfriend that he attacked her on Oct. 30.
The six-year veteran spent a night in jail before he was freed on $5,000 bond.
"You know what? Varly's been like this since the start of the year, and he's been very focused," Avalanche coach Patrick Roy said. "In some ways, it was not a big surprise. Those charges were dropped (three of the four), which is good news for him.
"He made a couple of big saves at the right moments. He's having the support of his teammates, and his teammates want to play hard in front of him, as well."
The Avalanche, who have allowed an NHL-low nine first-period goals, blocked nine shots during the first 20 minutes and finished with 15.
Los Angeles received a 4-minute power play with 77 seconds left in the second period, after John Mitchell cut Willie Mitchell with a high stick, but the Kings got only two shots during that stretch and finished 0 for 4 with the man advantage.
"Team defense is something we pride ourselves in, and we try to take away time and space" defenseman Nate Guenin said. "Our forwards and defensemen take pride in the defensive zone, and we've got two rock-solid goalies back there [including Jean-Sebastien Giguere]. So if we make a mistake, they are there to bail us out."
McGinn's fourth goal of the season, and first game-winner, helped the Avalanche (17-5-0) improve the best start in franchise history. They are 15-0 when they score first.
Ben Scrivens made 32 saves in his sixth straight start for the injured Jonathan Quick. He got a bad break when a rebound trickled into the net off the elbow of teammate Jarret Stoll after Scrivens made the initial save on McGinn.
"I'm just trying to drive to the net, get a lot of wood on it, and make sure I hit the net," McGinn said. "I thought Scrivens made a great save on the shoulder, and it was unlucky for them that it hit their guy -- but lucky for us that it hit the back of the net. We worked hard the whole game, so we're glad we got rewarded."
The Kings, coming off a 2-1 overtime loss to New Jersey on Thursday, were shut out for the first time this season and ended an eight-game stretch in which they recorded at least one point.
Scrivens was at his best during the second one, stopping point-blank shots by John Mitchell, Ryan O'Reilly, rookie Nathan MacKinnon and two more by P.A. Parenteau after Willie Mitchell was off for high-sticking Maxime Talbot with 7 minutes left in regulation.
"It's hard to win in this league, so you can't take anything for granted," Scrivens said. "You have to respect Colorado. They are a good team and they find ways to win. I don't think a lot of people at the start of the season had them where they are in the [standings], so you have to give a ton of credit to them."
The Kings activated LW Kyle Clifford from the injured list after he missed six games with an upper-body injury. C Trevor Lewis was placed on the injured list because of a lower-body injury. ... Colorado's Marc-Andre Cliche, who came in with the team's lowest winning percentage on faceoffs (34 percent) won seven of eight. ... The Avalanche are 5-4 since LW Alex Tanguay sustained a knee injury on Nov. 2 against Montreal. He is the only player left from Colorado's 2001 Stanley Cup-winning team. ... MacKinnon, the first overall pick in the June draft and the NHL's youngest player at 18, leads rookies with 65 shots. He has three goals. ... Colorado was 0 for 3 on the power play.