After Wild goalie Matt Hackett stopped Oliver Ekman-Larsson, Heatley put a backhand past Jason LaBarbera for the win. LaBarbera made 22 saves.
Hackett took over for Josh Harding, who left with a lower-body injury in the second period.
Kyle Brodziak and Steven Kampfer scored first-period goals for the Wild.
Down 2-0 in the third, Raffi Torres got the Coyotes on the board when he scored at 1:40 with his 11th goal of the season on a pass from defenseman Michael Stone.
Just 1:05 later, Lauri Korpikoski converted a pass from Antoine Vermette for his 15th goal of the season to tie the game.
The Wild won for the first time in six games. With the shootout loss, the Coyotes remain winless in their past five games.
Harding left after stopping a drive from Ekman-Larsson at 11:38 of the second period. Harding will evaluated on Friday.
He was replaced by Hackett, appearing in only his eighth game this season. Harding made 24 saves. Hackett stopped 15 of 17 shots.
The Coyotes got off to a slow start.
Defenseman Keith Yandle's pass through the middle was intercepted by Darroll Powe, whose 25-foot drive was tipped in front by Brodziak at 4:06 of the first period.
Kampfer, recalled from AHL Houston on March 3, scored his first goal of the season on a wrist shot from inside the left faceoff circle at 5:04.
The Coyotes outshot the Wild 32-15 through the first two periods, but could not score.
Notes: Stone's assist on Torres' goal was his first NHL point. . Phoenix's Martin Hanzal caught a high stick in the face late in the first period, and did not return.