Koivu beat Semyon Varlamov with a shot just under the crossbar. Zach Parise also scored for the Wild in the shootout.
Matt Duchene had a shootout goal for the Avs, but Harding then stopped P.A. Parenteau and Ryan O'Reilly to preserve the win.
Harding finished with 26 saves as the Wild won their first game away from Excel Energy Center since Nov. 23. Harding entered the game with a league-best 1.52 goals-against average.
Ryan Suter tied the game with 3:53 remaining in regulation.
Maxime Talbot deflected in a goal in the second period and Varlamov made26 saves.
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