Minnesota outshot the Oilers 18-0 in the second period but the game remained tied 1-1 entering the final frame.
Koivu, however, found the net just 9 seconds into the third and Charlie Coyle followed with the eventual game-winner 6 minutes later, giving the Wild the momentum they'd need to register their third win in four games.
Jared Spurgeon also tallied with a helper and Dany Heatley added a power-play marker in the victory.
Niklas Backstrom made 19 saves to record the win.
Magnus Paajarvi and Sam Gagner each found the net and Devan Dubnyk stopped 39 shots as the Oilers dropped their third in four outings.