Staal put the Hurricanes ahead for good at the 13:12 mark when he got loose in the crease and converted a pass from Patrick Dwyer.
Sutter followed 57 seconds later with a backhander for a 3-1 lead.
Dwyer finished with two assists and Anton Khoudobin turned aside 38-of 40 shots for Carolina, which won a third straight.
Thomas Vanek had a goal and an assist for the Islanders, whose own three-game winning streak was broken.
Evgeni Nabokov made 27 saves for New York.
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