Tavares cashed in New York's second power-play goal when he flipped a shot over Ward at 5:23 of the second period for his seventh goal.
Before the goal announcement could be completed, the Hurricanes tied it again. Skinner matched Tavares' goal total when Jordan Staal found him in the slot for a shot that beat DiPietro between the pads at 5:52.
But then the Islanders' suddenly potent power play clicked for the third time. Brad Boyes fed a cross-zone pass to Moulson, who was alone near the right post. He went down to a knee and pumped a shot past Ward at 7:46 for his fifth goal, but first at home.
New York survived three trips to the penalty box in the second period, including two in the opening 1:18.
Nielsen opened the scoring off a crisp feed in front by Josh Bailey, who recorded his first assist of the season in his second game. Bailey quickly moved the puck from the boards in front to Nielsen, who slid a shot past Ward at 2:48 of the first.
That was the first man-advantage tally against Carolina in four games following 11 straight kills.
Ward was otherwise strong in the period in making 11 saves. Carolina generated only six shots at the other end, but got one to pinball off sticks and a skate to get even.
Jamie McBain fired a drive from the top of the right circle that hit Islanders forward David Ullstrom in the circle on its way to the net. The puck then struck Wallace's skate and was redirected into the net for his first of the season at 13:42. A brief video review was held before the goal was confirmed.
"There were some crazy bounces," DiPietro said.
NOTES: The last time the Hurricanes allowed four power-play goals and won was Oct. 28, 2005, against Philadelphia. New York hadn't scored four power-play goals and lost since Oct. 14, 1978 — 10-7 at Toronto. ... The Islanders are 4-2-1 when they score a power-play goal. ... Nabokov had started seven straight games. ... McBain was a healthy scratch in Carolina's loss at Philadelphia on Saturday. ... Wallace's past two goals have been scored against the Islanders. ... Lubomir Visnovsky earned an assist on Moulson's second goal for his first point with the Islanders in his second game.