Jason LaBarbera, making his first start for Phoenix since November 10, turned aside 33 shots.
Down 2-0, Cullen crashed the crease and lifted a rebound over LaBarbera to put Minnesota on the board at 12:13 of the third.
Upshall restored the Coyotes’ two-goal advantage with 2:40 remaining, but Koivu scored off a nifty Martin Havlat feed from behind the net to pull the Wild within one with 66 seconds left.
Korpikoski’s empty-net goal inside the final minute sealed the win.
Jovanovski tipped an Adrian Aucoin point shot past Theodore to give the Coyotes a 1-0 lead with 53.5 seconds left in the first period.
Phoenix led by two after Upshall one-timed Shane Doan’s cross-ice pass at the 5:43 mark of the middle stanza.
The Coyotes swept the four-game series from Minnesota last season, outscoring the Wild by a 14-8 margin…Phoenix improved to 6-0-0 against teams from the Northwest Division…Forward Pierre-Marc Bouchard returned to the Minnesota lineup. He had missed 104 straight games and 112-of-113 games overall since suffering a concussion against the New York Islanders on March 25, 2009.