Anaheim faced a 4-2 deficit heading into the third period, but Jason Blake brought the Ducks within one on a power-play goal midway through the stanza.
The Ducks tied it with 32 seconds left in regulation on a wrist shot by Lubomir Visnovsky.
Koivu then intercepted a pass by Bruno Gervais and fired a wrist shot that beat Martin Biron for the game-winner.
Anaheim has won three in a row, but is still seven points out of the final playoff spot in the Western Conference.
Jonas Hiller allowed three goals on 17 shots and was removed from the game midway through the second period. Curtis McElhinney came on to stop 13 shots and get the win.
Anaheim's Teemu Selanne remained stuck on 599 career goals. He is trying to become the 18th player in NHL history to reach 600.