The score was tied 1-1 3 minutes into the final period when Eric Staal won a face-off to the left of the Coyotes' net. Skinner snared the puck in the slot and shot it in one motion, beating goalie Mike Smith.
Staal later added an empty-netter and Nathan Gerbe also scored as the Hurricanes snapped a three-game skid.
Justin Peters earned the decision with 37 saves.
Paul Bissonnette scored and Smith yielded two goals on 32 shots for Phoenix.