Korpikoski's second goal of the game 15 seconds into the third period lifted the Coyotes into a 3-3 tie.
Following a scoreless overtime, the game went to shootout.
Sullivan scored while Smith turned aside Nick Bonino, Corey Perry and Koivu to hand the Coyotes a victory in the first of three straight games against Pacific Division-leading Anaheim.
Smith finished with 31 saves in the victory.
Andrew Cogliano supplied a hat trick while Koivu finished with a goal and a helper for the Ducks.
Viktor Fasth turned aside 33-of-37 shots in the loss.