Nelson completed a hat trick when he converted Mathew Barzal's fifth assist of the game with 1:45 remaining in overtime.
[WATCH: All Red Wings vs. Islanders highlights]
Nelson, Anders Lee, Nick Leddy and Josh Bailey scored 3:37 apart on the power play to make it 6-5 for New York (27-23-6), which rallied from trailing 5-2 after Tyler Bertuzzi was given a match penalty for slashing Cal Clutterbuck. The four goals on the same power play are an Islanders record.
Video: DET@NYI: Leddy buries slap shot for game-tying PPG
The win moved New York into the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
Nelson, who is the Islanders' third center behind John Tavares and Barzal, entered with two goals in his previous 25 games.
Of wanting to produce like the Islanders' top six forwards, Nelson said, "Trying to … I may be a little late. I'm trying to chip in, just help the team win in any way. We need all the points we can get right now."
Thomas Greiss allowed five goals on 32 shots for New York before being pulled in the third period. Jaroslav Halak made six saves.
"That was a roller-coaster of a game," Barzal said. "So much emotion."
Henrik Zetterberg had a goal and three assists for Detroit (21-23-9), which is nine points behind New York. Mike Green sent the game to overtime when he scored with 29 seconds remaining in the third period.
Video: DET@NYI: Green ties it late on redirection
"We can control one thing, that's our actions and our attitude," Red Wings coach Jeff Blashill said. "We can't take a penalty at that point in the game. We can't take a retaliatory penalty. It was unnecessary.
"The game's over. We've taken their life away, and then we gave them life."
Anthony Mantha and Martin Frk scored during a three-goal first period for the Red Wings, who had scored two goals or fewer in nine of their previous 12 games. Petr Mrazek made 22 saves.
"Every point is important, and losing one tonight like that hurts," Mantha said.
Nelson and Lee got the Islanders within one with power-play goals after Bertuzzi was given the five-minute penalty at 13:27 of the third. Nelson made it 5-3 at 14:34 before Lee scored his 29th at 15:28.
Video: DET@NYI: Lee cashes in on loose puck for PPG
Leddy tied it 5-5 with 3:21 remaining when his shot found its way through a crowd in front and past Mrazek. Bailey gave the Islanders a 6-5 lead when he one-timed a pass from Tavares at the right face-off circle at 18:11.
"Hopefully this is a date that you circle," Islanders coach Doug Weight said.
Jordan Eberle scored at 9:36 of the second period when he took a pass from Barzal and beat Mrazek with a wrist shot to make it 3-1.
Nelson made it 3-2 at 3:02 of the third period when he redirected Thomas Hickey's shot from the right point.
Darren Helm scored at 7:42 of the third period when Frans Nielsen's harmless wrist shot from the left wing trickled between Greiss' skates, and Helm was at the far post to make it 4-2.
Niklas Kronwall's power-play goal made it 5-2 at 9:54 of the third. Greiss was pulled for Halak following the goal.
Greiss has allowed at least five goals in nine of 21 starts this season.
"The fourth goal obviously wasn't good," Weight said. "I thought Tommy was pretty good after that third goal. He beared down. I wasn't happy with the fourth one and the fifth one I thought went straight in, and that's frankly why I pulled him. I just saw on the tape that it was tipped straight up.
"It's a win, and he contributed. We've got to keep him going, keep him positive."
Mantha gave the Red Wings a 1-0 lead 2:13 into the first period, and Frk scored his 10th of the season at 5:12.
Video: DET@NYI: Mantha goes five-hole on breakaway
Zetterberg made it 3-0 at 14:40 when he put a rebound past Greiss.
"We've got to find a way to stop playing from behind," Bailey said. "It's good to have [our offense] in our back pocket, but I think we're a better team when we're the ones controlling the pace and making the other team chase the game a little bit."
Goal of the game
Nelson's goal at 3:15 of overtime.
Video: DET@NYI: Nelson finishes hat trick with OT winner
Save of the game
Halak's save against Mantha at 1:40 of overtime.
Video: DET@NYI: Eberle helps out Halak in OT
Highlight of the game
Bailey's power-play goal at 18:11 of the third period.
Video: DET@NYI: Bailey caps Islanders' PPG barrage
They said it
"It was more or less sticking up for myself, not taking stuff from anyone. I regret it. At the end of the day I don't think it was a five-minute penalty." -- Red Wings forward Tyler Bertuzzi
"I don't think I've ever been a part of one like that. That was a pretty wild finish. Just the ups and downs of a game and a season kind of all wrapped into about 15 minutes there. Good one to win, though." -- Islanders forward Josh Bailey
Need to know
It was Barzal's third five-point game of the season. He is the second rookie in NHL history, and the first since Joe Malone of the Montreal Canadiens in 1917-18, to have three five-point games. … The Islanders are the third team in NHL history to trail by at least three goals in the final 5:26 of the third period and win. The Minnesota North Stars did it in a 7-6 win against the Canadiens on March 12, 1984, and the Canadiens did it in a 5-4 win against the Ottawa Senators on March 15, 2014. … New York defenseman Johnny Boychuk returned after missing 18 games because of a lower-body injury.
Red Wings: At the Washington Capitals on Sunday (3 p.m. ET; SN1, SNE, SNO, SNW, TVAS, NBCSWA, FS-D, NHL.TV)
Islanders: Host the Calgary Flames on Sunday (7 p.m. ET; SN, MSG+, NHL.TV)
Video: Nelson caps hat trick in OT as Isles beat Wings, 7-6