Koivu scored on a backhander and Ebbet slipped the game-winner past Colorado goaltender Craig Anderson in the second round.
Guillaume Latendresse and Eric Belanger scored in regulation for Minnesota. Goalie Josh Harding had 20 saves for his first win of the season. He turned in a clutch skate save on Wojtek Wolski in the shootout.
Matt Hendricks and Matt Duchene scored for the Avalanche. Anderson made 34 saves.
Colorado led 2-0 at the end of the first period. Minnesota responded with a goal 57 seconds into the second period by Belanger. Latendresse tied the game on a rebound in the slot with 8:46 left in regulation.