Varlamov had 27 saves in regulation and overtime, and Daniel Briere scored in regulation for Colorado. It was the fifth straight shootout win for the Avalanche, who are 6-2-2 in their last 10.
''It's great to start the new year, with a big win,'' Duchene said. ''We want to roll through January here.''
He got his team off to a good start.
After both teams failed to convert on their first shootout opportunities, Duchene skated slowly into the zone, got goalie Ben Scrivens to go down and beat him with a low shot.
''Me and Mario (Duhamel), our video coach, come up with something before each game,'' Duchene said. ''I think the biggest thing is each goalie has a different style and I just tried to get him to bite and make him change direction.''
Varlamov made a save on Jordan Eberle, and Tanguay skated slowly at the net and finished it with a roof shot over a sprawled Scrivens.
''It's tough when they come in so slow,'' Scrivens said. ''It's not a game situation where they have back pressure. Duchene has a good shot. You have to respect his shot. He is a good player.''
Ben Pouliot scored for Edmonton, and Scrivens finished with 27 saves. The Oilers, who are last in the NHL in points with 25, are 1-7-4 in their last 11 games and 1-9 in games decided after regulation.
''I thought our overtime period was a lot better than we've seen in the past,'' Edmonton coach Todd Nelson said. ''We had puck control, we had some good, quality chances, but we have to shoot the puck in those situations.''
Neither team generated much offense until the Avalanche broke through in the second period. Duchene, one of the many Colorado forwards struggling to score, carried the puck behind the net to the front and fed Briere on the opposite side. Briere knocked the puck into an open net at 7:55 of the period.
It was his seventh of the season and first in 13 games.
''I was able to get a step on the guy coming out front,'' Duchene said. ''I was just trying to put it over there. He is such a smart player and wouldn't miss that one.''
Edmonton generated just five shots in the second period, but gained some momentum with three consecutive power plays. After failing to capitalize on the first two, Pouliot tied it at 4:47 of the third with his sixth of the season. The big left wing, who returned to action after missing 18 games with a broken foot, muscled through defenseman Jan Hejda to knock a rebound past Varlamov to tie it at 1.
''He made an impact for us,'' Nelson said of Pouliot. ''He played a simple, solid game and capitalized on the goal, obviously. This is kind of a tough spot for him, coming back here. It's hard to breathe, but he had short shifts.''
Both teams had chances to take the lead. Colorado's Gabriel Landeskog's backhander at a wide-open goal hit the side of the net, and Edmonton's Teddy Purcell hit the right post with a shot midway through the third.
Colorado didn't get its first shot on net in the third until 7:15 was left in regulation. The Avalanche were outshot 12-3 in the third.
NOTES: The Oilers acquired C Rob Klinkhammer and a 2015 first-round pick from Pittsburgh for LW David Perron. ... Edmonton D Keith Aulie served the first game of his two-game suspension for his hit on Calgary F Matt Stajan on Wednesday. ... Avalanche D Zach Redmond was a healthy scratch for the second straight game.