Marek Svatos, Brad Richardson and Joe Sakic scored in regulation for the Avs, while Ales Hemsky, Petr Sykora and Joffrey Lupul countered for Edmonton.
It was also the first win for the Oilers (31-37-7) in the 12 games since their deadline deal that sent forward Ryan Smyth to the New York Islanders.
The defeat snapped a four-game winning streak for the Avalanche (38-29-7), but they remained undefeated in regulation in their last 11, improving to five points behind Calgary for the final playoff spot in the Western Conference.
Colorado started off the scoring nine minutes into the first period as Ian Laperriere found Richardson open at the left faceoff circle. With a pass from behind the net, Richardson blasted a one-timer past Oilers starter Dwayne Roloson for a 1-0 Avs lead.
Colorado out-shot Edmonton 14-5 in the opening 20 minutes of play.
Colorado added to its lead early in the second, effectively passing the puck around on the power play before Sakic blasted his 32nd of the season past Roloson to extend his scoring streak to five games.
Edmonton cut the lead in half seven minutes later as Hemsky, back after an eight game absence, paid immediate dividends, catching Avs netminder Peter Budaj off guard, as he skated around the back of the net and banked a wraparound shot off Budaj's skate for his 50th point of the season.
The Oilers pulled even five minutes later as Sykora fired a power-play shot from the high slot over Budaj's shoulder for his first goal in 15 games, tying the contest 2-2.
Colorado pressed hard to regain the lead with a power play late in the period, but Roloson stood on his head, making several big saves to keep the score even after the middle frame.
The Oilers' netminder wasn't nearly as sharp on the Avalanche's third goal 4:40 into the third, as he was slow to clear a puck from the crease on a Colorado power play and Svatos swatted it through his legs for a 3-2 Avs lead.
Colorado looked to have the game in hand but a turnover changed that as Lupul came away with a puck in the Avalanche zone and notched his 16th of the season with just 1:47 left to play, sending the game to OT and eventually the shootout.
The Oilers host Nashville on Saturday, their final home date of the season. The Avalanche finish off a three game trip in Vancouver on Sunday.
Notes: The injury-plagued Oilers finally got some relief on Friday as forwards Ales Hemsky and Marty Reasoner and defenceman Tom Gilbert were all able to return to the lineup, cutting the number of injured Edmonton players to seven ... With three healthy bodies back Edmonton had to send three of their emergency call-ups back down, shipping forwards Robert Nilsson and Kyle Brodziak to Wilkes-Barre of the AHL and blue-liner Sebastien Bisaillon to Val d'Or of the Quebec Major Junior League. ... Petr Sykora is the only player on Edmonton's current roster with more than 20 goals this season. Colorado has five ... With five goals and six assists in his last five games, Joe Sakic is now only two points shy of becoming just the fifth player in NHL history to hit the 90-point mark at 37-years or older ... The Avalanche have eight games remaining as they try to catch Calgary for the final playoff spot in the Western Conference, including two against the Flames. They also play Vancouver three times and have one game stints against Phoenix, Minnesota and Nashville.