Pierre-Marc Bouchard had the other goal while Niklas Backstrom made 27 saves for the Wild Gaborik's second goal of the game with 4:34 remaining was the winner. Koivu added an empty-net goal in the final minute. The Wild took the lead three times through the first two periods only to have the Avalanche counter.
Bouchard and Gaborik scored in the second period, but Colorado answered with goals from Brett Clark and Marek Svatos.
Andrew Brunette's first-period goal off a Minnesota turnover tied it after Koivu's goal staked the Wild to an early lead.
Milan Hejduk had two assists for Colorado, and Peter Budaj stopped 32 shots.