The Coyotes won for the first time this season after dropping their opening two contests. It was Sullivan's first hat trick in more than three years.
Sullivan tied the game midway through the second period and Oliver Ekman-Larsson put the Coyotes in front for good with 4:38 to go in the middle stanza.
Phoenix outshot Columbus during the third period 10-8 with Sullivan scoring the Coyotes' first two goals of the frame. Ekman-Larsson added his second goal with 4:58 to play.
All five Phoenix goals came against Steve Mason, who faced 42 shots.