Nielsen scored three times in the third period to lead the Islanders to a 7-5 victory in the Stars' final visit to Nassau Coliseum.
The Stars kept Islanders captain John Tavares off the scoresheet, but allowed seven goals for the second time in as many visits to the Coliseum in 2014. Tavares had a hat trick in a 7-3 win on Jan. 6.
Nielsen, who came into the game with one goal in New York's first seven games, broke a 4-4 tie at 9:39 of the third period when he finished off a 2-on-1 break with Nikolay Kulemin by ripping a shot from the slot past Dallas goalie Anders Lindback. He made it 6-4 with 4:51 remaining by beating Lindback on another 2-on-1, this time with Mikhail Grabovski, and completed the hat trick by scoring into an empty net with :00.1 seconds remaining.
He didn't celebrate his first NHL hat trick right away because he wasn't sure the puck had crossed the goal line before time expired.
"I couldn't really celebrate because I didn't know if the last one counted," Nielsen said. "It's a pretty cool feeling."
The line of Nielsen, Kulemin and Grabovski, who was playing his first game after missing time with a concussion, made the Stars pay for some sloppy defensive play.
"They're really smart players," Nielsen said of Grabovski and Kulemin. "They compete really hard. I've played against them. They are frustratingly hard to play against, so it's fun playing with them now."
Kulemin, Anders Lee, Brock Nelson and Lubomir Visnovsky also scored for the Islanders (6-2-0), who have won back-to-back games after losing two straight. Chad Johnson survived a shaky start and got the win with 26 saves.
"We know we're a good team and we're working hard," Nielsen said of the Islanders, who are off to their best start since going 7-0-1 to begin the 2001-02 season. "We wanted to make a statement in the beginning, and we've done it."
"We created more than enough chances, but we didn't adjust well enough," Dallas coach Lindy Ruff said. "We did some good things and played hard, but we were inconsistent."
It was a difficult night for Lindback, who hadn't played since Oct. 11 and didn't get a lot of defensive help from his teammates.
"It was quite a while between games, and he had a tough night," Ruff said of Lindback. "We gave [the Islanders] some pretty good opportunities, but we've got to get some saves."
Lee, a rookie playing his second game this season after being recalled from Bridgeport of the American Hockey League, gave the Islanders the lead 5:11 into the game when he deflected Thomas Hickey's left-point shot between Lindback's legs.
Dallas tied it at 13:25 on a shot that Johnson thought he had stopped. Patrick Eaves fired a wide-angled shot from the left circle that Johnson thought he had covered. But the puck trickled past Johnson, and Fiddler stuffed it into the net for his first of the season.
New York went back in front at 18:42 with its first shorthanded goal of the season. Kulemin, who joined the Islanders as a free agent in the offseason, forced a turnover by Dallas defenseman Alex Goligoski, raced into the right circle and snapped a shot past Lindback for his first goal with the Islanders.
The Islanders took a 3-1 lead at 1:25 of the second period. Lindback stopped Ryan Strome's wrist shot through traffic from the high slot, but Nelson cleaned up the rebound for his team-high fifth goal.
The Stars got even with two goals in 85 seconds. Seguin scored his sixth of the season by snapping a shot from just above the right faceoff dot over Johnson's glove at 5:25, and Spezza tied it with a power-play slap shot from the right circle through a screen. It was his second goal.
Roussel gave the Stars their first lead of the night when he scored at 15:50, but Visnovsky tied the game again 30 seconds later when he slapped a rolling puck past Lindback from inside the right circle for his first goal of the season.
"We need that mental durability," Islanders coach Jack Capuano said of his team quickly responding after the Stars took the lead. "I'm seeing that game in and game out with these players."
Material from team media was used in this report.