Varlamov made seven stops in the third period as the Avalanche used a wrist shot by Ryan O'Reilly with 5:43 remaining to forge a 1-1 tie and force extra time.
After a scoreless overtime, Jarrett Stoll missed on the Kings' first shootout try and Varlamov came up with saves on Anze Kopitar and Dustin Brown.
That allowed Milan Hejduk's first-round tally to stand up as the game-winner for red-hot Colorado, which took its ninth win in 10 games.
Brown got the only regulation goal as the Kings fell for the first time since Darryl Sutter took over as coach Dec. 22, going 4-1-2 in the span.
Jonathan Quick made 20 saves for Los Angeles.
Notable deaths of 2014 [PHOTOS]
Julius Randle breaks leg during NBA debut