Minnesota won for the third time in four games, taking the lead for good on goals less than 2 minutes apart from Devin Setoguchi and Jonas Brodein in the second period.
Those scores gave the Wild a 3-1 lead, but they did not clinch the contest until Koivu tallied into an empty net with 34 seconds remaining. Koivu also scored in the second period to give Minnesota a 4-2 advantage.
Niklas Backstrom made 29 saves in the victory.
P.A. Parenteau scored twice for the Avalanche. Colorado has gone 0-5-3 in its last eight road contests. The Avalanche will host Minnesota in a rematch Saturday.