Vincent Trocheck tied it 1-1 with 57 seconds remaining with the Carolina net empty and the extra attacker.
"It hurts whenever you get one back late and they score," Trocheck said. "But it starts at the beginning of the game. We've got to start better. I don't know what it is, but if we get off to a better start in the first period and we get out to a lead, it's a different game."
Jean-Gabriel Pageau scored, and Palmieri had an assist for the Islanders (33-28-9), who have won five of six. Ilya Sorokin made 20 saves after missing five games with an upper-body injury.
Frederik Andersen made 19 saves for the Hurricanes (46-18-8), who have lost three of four and remain two points ahead of the New York Rangers for first place in the Metropolitan Division. Carolina clinched a berth in the 2022 Stanley Cup Playoffs with a 5-3 win against the Buffalo Sabres on Thursday and was playing its third game in four days.
"We were really gassed," Hurricanes coach Rod Brind'Amour said. "There was no energy in our legs, so we had no speed to get in and get on them. [The Islanders] did a nice job with a lot of blocked shots and did that kind of thing to keep us away from getting any real traction after the first period."
The Islanders took a 1-0 lead at 8:24 of the second period after Palmieri forced a turnover behind the net and Martin fed Pageau alone near the top of the crease. It was Pageau's fifth goal in three games.
"'Marty,' we all know more of his physical game, but I think his skill is very underrated," Pageau said. "He made a really good pass there. He was very composed with the puck, patient to wait for the lane to open. He made me a perfect pass. All I had to do was put it in."