John Tavares had two goals and an assist to 3, at Nassau Coliseum.
Kevin Poulin stopped 33-of-36 shots to earn the win.
"Overall, everyone did a great job of battling through. It probably wasn't our best game but we found a way," said Tavares. "We turned it up when we had to and made some plays in key moments of the game."
Cam Ward allowed three goals on 21 shots through two periods, but did not return for the third because of a lower body injury. Justin Peters stopped 9- of-10 shots in relief.
The contest entered the final period tied at three goals apiece, but it took the Islanders just 38 seconds to get the eventual game-winning goal past the newly-entered Peters.
Matt Moulson received a pass in the slot in an offensive rush and ripped a wrister at Peters. The goaltender made the initial save and then a second on Moulson after he gathered his own rebound, but Tavares was there to clean up that rebound from the left side with a backhander into the twine.
"They scored the majority of their goals on the rush," said Hurricanes forward Eric Staal. "And that's where they hurt us."
Poulin made 12 saves in the third period to preserve the win, including several in an offensive flurry by Carolina in the closing moments that saw Jeff Skinner fan on an open-net opportunity within the final five seconds.
The Islanders struck first 2:07 into the first period with Okposo's hard wrister from the high slot that sailed past Ward.
Carolina, though, sent the game into the first intermission tied at one when Sutter finished off a 2-on-1 rush with Patrick Dwyer by lifting a backhand past Poulin's stick side with 6:13 left in the opening frame.
Tlusty gave the Hurricanes the lead just 15 seconds into the second period by redirecting a Jerome Samson pass from the right circle into the net.
But back-to-back goals by Tavares and Nielsen within a four-minute span gave the Islanders a 3-2 edge.
Tavares' tally came at the 4:10 mark when he swept down the right side of the ice before cutting across the goal and sliding a shot past Ward, and Nielsen's marker came on a power play at the 8:01 mark when he finished off a tic-tac- toe play with a one-timer from the right circle.
However, Spacek knotted the game at three with 9:02 left in the middle stanza with a slap shot from the left point that went through a crowd and into the short side after Staal won an offensive zone faceoff.
New York swept the season series with Carolina, 4-0...The Islanders went 1- for-2 on the power play, while the Hurricanes went 0-for-3 on the man advantage...Mark Streit also had two assists for the Islanders.