Ilya Sorokin needed only 19 saves to pick up the victory, though he was less than a minute away from matching a franchise record for shutouts in a season.
But Vincent Trocheck’s goal with 56.8 seconds to play tied it for Carolina.
Still, the Islanders won for the fifth time in their last six games — thanks to Palmieri’s clutch goal.
Jean-Gabriel Pageau also scored for New York, and Palmieri had an assist on that goal.
A night after clinching a playoff spot for the fourth year in a row, the Hurricanes weren’t crisp at all on their way to losing for the third time in four games. Frederik Andersen made 19 saves.
Trocheck’s tying goal came after the Hurricanes pulled Andersen for an extra skater. Trocheck didn’t have a goal or assist in the previous five games.
Sorokin appeared for the first time in New York’s last six games, with his previous outing coming March 27. He wasn’t threatened during stretches by the Metropolitan Division’s highest-scoring team.
The Islanders won the battle for the puck behind the net before Matt Martin delivered a pass to Pageau, who put the puck in the upper part of the net at 8:24 of the second period. Pageau has five of his 15 goals this season in the last three games.
The Hurricanes managed only one shot on goal in the second period. They failed to generate a scoring chance on a power play late in the period.
Carolina’s Martin Necas had a first-period opportunity when Sorokin left an opening, but Necas’ shot on the power play bounced off the goalie’s pad.
This was the first meeting between the teams since Carolina won 6-3 at home to open the season.
Sorokin has six shutouts this season, one shy of the franchise record set by Chico Resch (1975-76) and matched by current teammate Semyon Varlamov (last season).
Sorokin, who played in his 66th NHL game, has nine career shutouts.
DELIGHT IN DIVISION
The Islanders sit in fifth place in the Metropolitan Division, but they have won 10 of their last 12 games against members of the division. New York’s problem was winning just twice in divisional matchups in its first eight attempts.
Carolina remained at 100 points (46-18-8) for the second-most in franchise history. With 10 games to go, the Hurricanes need to secure 12 points to match the franchise record set in the 2005-06 season.
Islanders: At St. Louis on Saturday night.
Hurricanes: Host Anaheim on Sunday night.