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.
Nino Niederreiter tied the game with 3:53 remaining in regulation.
Maxime Talbot deflected in a goal in the second period and it appeared as if Varlamov might make that hold up. Varlamov finished with 26 saves.
Neither team could muster many scoring chances in overtime.
Midway through the second period, Cory Sarich unleashed a shot from just inside the blue line that deflected off of Talbot's skate and past Harding. It was Talbot's second goal for the Avs since being acquired from Philadelphia on Oct. 31.
Niederreiter tied it up when he deflected Ryan Suter's shot in front of the net and it trickled past Varlamov.
Once again, the Avalanche struggled on the power play. They're now 0 for 26 over their last 10 games.
It was a rather clean first period, with no goals and just one penalty on Minnesota's Jonas Brodin for interference.
Harding came up big in the opening 20 minutes and stopped all 11 shots he faced, including a point-blank attempt from Cody McLeod.
Defenseman Jan Hejda was back on the ice for Colorado after missing a week with a right knee injury. He had to be helped off the ice in Edmonton on Dec. 5 after losing an edge and going feet first heavily into the boards.
"Wasn't as bad as how it looked on television," Hejda joked before the game.
NOTES: The Wild juggled around their lines after recalling C Erik Haula from Iowa of the American Hockey League on Friday. ... The Avalanche held a moment of silence before the game for Arapahoe High School staff, students and families who were affected by the shooting at the suburban Denver school on Friday. ... Avalanche D Ryan Wilson (back) missed his 17th straight game, but may be moving closer to a return. "We want him to have a couple of practices before he plays," coach Patrick Roy said. ... F John Mitchell played in his 300th game. "It's a good number," said Mitchell, who had an assist on Talbot's goal.