Peter Regin added a goal with a helper while Colin McDonald and Matt Donovan also tallied for the Islanders, who scored three times in the second period to blow the game open.
Evgeni Nabokov stopped 33-of-34 shots to record the victory.
Coyotes goalie Mike Smith was pulled after two periods, during which he allowed five goals on 20 shots. Thomas Greiss finished up in net with seven saves on eight chances.
Martin Hanzal got the only goal goal for Phoenix, which fell to 1-2-0.