Joe Pavelski located the loose puck during a net-from scrum and sent a backhand pass to Duchene for a shot over a sprawling Alexandar Georgiev.
Mason Marchment nearly won it for the Stars with an apparent goal at 12:31 of the first overtime on a wrist shot from the slot, but it was immediately waved off because of goaltender interference against Duchene. The call was upheld following video review.
Dallas will face either the Vancouver Canucks or Edmonton Oilers in the Western Conference Final. Vancouver leads that series 3-2.
Jamie Benn scored the tying goal early in the third period, and Jake Oettinger made 29 saves for the Stars, who are the No. 1 seed from the Central Division.
Mikko Rantanen scored, and Georgiev made 36 saves for the Avalanche, who are the No. 3 seed from the Central.
Rantanen gave Colorado a 1-0 lead at 5:48 of the second period on the power play. He roofed a wrist shot short side over the shoulder of Oettinger from below the right face-off circle.
Benn tied it 1-1 at 1:56 of the third, receiving a behind-the-back pass from Evgenii Dadonov in stride and stick-handling past a sprawling Georgiev before releasing a backhand shot into the open net.
