Wojtek Wolski and Milan Hejduk also scored in the shootout for Colorado, which pulled even with Calgary for sixth in the Western Conference playoff race with 90 points.
Tom Gilbert had a goal and an assist, and Dwyane Roloson made 32 saves for the Oilers, who, with 84 points, are two points behind Nashville. The Predators passed Vancouver and moved into eighth place in the West.
The Avalanche pulled Jose Theodore with 1:15 left in regulation and kept the play in the Edmonton zone. Peter Forsberg's pass from behind the net went in front of the goal, and Sakic roofed a shot for his 12th of the season.
Andrew Brunette posted two goals and an assist, Forsberg had three assists, and Theodore had 27 saves for Colorado, which has three games left.
All four Edmonton scorers notched their 13th goals of the season.
Sam Gagner gave Edmonton a 4-3 lead with a shot from a sharp right angle that beat Theodore with 2:57 left in the second period.
Curtis Glencross made it 3-2 when he scored from the left boards with 12:06 left in the second. Brunette tied it with his 19th less than 4 minutes later when Forsberg fed him from behind the net.
In the first period, the Oilers killed two penalties, including a 5-on-3 power play that lasted 1:19. As the second penalty expired, Brunette knocked in Ryan Smyth's shot that had hit the post and gave Colorado a 1-0 lead with 12:38 left in the first.
Just 40 seconds later, Hejduk scored his team-leading 27th on a bang-bang pass play from Paul Stastny and Forsberg to make it 2-0.
