Palmieri cut it to 2-1 at 10:48 when he circled the net and scored through the five-hole from low in the right circle.
The Islanders tied it 2-2 with 21 seconds remaining in the period. Adam Pelech intercepted a pass from Brady Skjei in the neutral zone and pushed the puck ahead to Mathew Barzal, who scored on a glove-side wrist shot from above the slot.
Nelson gave New York a 3-2 lead at 9:18 of the third period when he took a pass from Palmieri in the neutral zone and slipped past Carolina forward Martin Necas before scoring on a wrist shot from the left circle.
"As a goalie, you let those goals in, especially now after the game, you can look at those and think where you can do better," Raanta said. "The win is the only thing that matters. They guys played great in front of me."
Slavin tied it 3-3 at 12:19 when he pinched down to the goal line on the left side and scored off Sorokin's mask inside the near post.
"Honestly, I was looking for the pass the whole time," Slavin said, "then [at the] last second, I saw his head and put it in the vicinity and ended up getting a lucky bounce."
Despite losing the first two games, the Islanders said they feel good about their chances heading home.
"Two fluky goals that usually don't go in," New York forward Zach Parise said. "We played well enough to win, and we'll move forward and do our best to grab the third one.
"You look at the good things. We were in the driver's seat. We were in a good position to win the game. It didn't happen, but I think if you look at the game as a whole we'll leave there saying to ourselves that there's a lot of really good things that we did and just a couple fluky goals. That's the difference."
NOTES: Hurricanes goalie Frederik Andersen did not dress because of an illness. He was scheduled to be the backup; it was Pyotr Kochetkov instead. … The Hurricanes/Hartford Whalers franchise is 28-17 in overtime Stanley Cup Playoff games. ... Brind'Amour got his 26th Stanley Cup Playoff win, passing Paul Maurice for the most in franchise history. … In addition to Teravainen, Carolina is without forwards Andrei Svechnikov and Max Pacioretty. Each has a torn ACL.