Cody McLeod and Mueller added goals in regulation for the surging Avalanche, who kept within striking distance of the eighth and final playoff spot.
Ryan Smyth and Taylor Hall scored for the Oilers.
Varlamov remains hot in shootouts this season as he improved to 7-0. He's also allowed just two goals 21 attempts.
The young Russian finished the game with 28 saves. But his biggest stops were of Jordan Eberle and Sam Gagner in the shootout.
Hejduk led off the shootout by sending a shot over the shoulder of Nikolai Khabibulin. Mueller soon followed with a similar shot.
The Oilers had the best scoring chance in overtime on a breakaway, only to have Varlamov stop a wrist shot from Ales Hemsky.
Colorado caught a break midway through the third period, when Hemsky's goal was waved off after Sam Gagner made incidental contact with Varlamov near the net.
Edmonton players were livid over the ruling.
At 15:17 of the third, the Oilers scored the equalizer as Hall knocked in a rebound underneath Varlamov.
Colorado nearly answered with 2:46 left as Paul Stastny's shot hit the post just before Khabibulin stopped it. But after a short review, the referees ruled the puck never crossed the line.
This is a pivotal stretch for the Avalanche, who have one more home game before venturing out on the road for three straight. They entered the day in the thick of the Western Conference postseason chase, a point back for the last spot.
The game featured a matchup of two of the top candidates for the Calder Trophy, which is given to the league's top rookie.
Ryan Nugent-Hopkins and Gabriel Landeskog were taken with the first and second picks, respectively, last summer and have lived up to the lofty expectations.
Landeskog ranks second among first-year players in points (43) and goals (18). He's also the only NHL rookie to lead his team in goals.
Despite missing 12 games with a shoulder injury, Nugent-Hopkins has 15 goals and 25 assists.
Mueller was the beneficiary of Hejduk's near-goal midway through the second period. Hejduk skated in on Khabibulin and tried to sneak in a shot with his forehand, only to reverse his field and attempt a backhand. Khabibulin stopped the shot, but slid out of position.
With the puck spinning on the doorstep of the crease, Mueller simply nudged it across the line.
Known more for his gritty play than goal scoring, McLeod tied the game at 1 early in the second period. He was knocked to his knees in front of the net, but sprung back up in time to receive a pass from Ryan Wilson, which McLeod lined past Khabibulin. Although McLeod has just six goals on the season, he has two over his past three games.
The Oilers had several scoring chances early in the game, but Shawn Horcoff's wrist shot clanged off the right post and Smyth couldn't quite stuff in a shot near the goal a few minutes later.
With 36.1 seconds remaining in the opening period, Smyth, a former Avs player, managed to put Edmonton on the board when he poked in a loose puck past a sprawled-out Varlamov. It was Smyth's 19th goal of the season.
NOTES: Avs F Matt Duchene (right ankle) is expected to be sidelined 3 to 4 weeks. ... Colorado F Chuck Kobasew missed his sixth game with a back ailment. ... Oilers F Ben Eager (flu) sat out a second straight game. ... Varlamov is 6-2-0 in his past eight starts.
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