After a scoreless overtime, the game moved to a shootout.
After Harding stopped P.A. Parenteau, Koivu sent a backhander under the crossbar for another score.
Harding then ended the game with a save on Ryan O'Reilly as Minnesota salvaged one victory on a three-game road trip.
Harding finished with 26 saves.
Maxime Talbot scored in regulation and Varlamov ended with 26 stops as the Avalanche fell for the third time in four games.