Nelson scored on the rebound of a shot from the left circle by Michael Grabner. Goalie Pekka Rinne made the save on Grabner's shot, but the puck kicked into the slot, hopped over the stick of Nashville defenseman Seth Jones and came to Nelson, who beat Rinne for his 18th goal of the season.
Nashville forward Craig Smith tied the game 3-3 at 10:47 on a wrist shot from point-blank range that beat goaltender Jaroslav Halak. Smith took a pass from rookie Filip Forsberg and lifted the puck over Halak's pads for his 21st goal.
Smith's goal came 31 seconds after Nashville's Mike Ribeiro scored to make it 3-2.
"They scored two quick so it might have deflated us a bit, but at the same time we knew it was just a tie game so you can't get too down on ourselves," Nelson said. "We went back out there, did a couple good things, found the goal and then from there on in the last six minutes or so we did a good job. Didn't give them much, and when they had the goalie pulled we kept them to the outside and did a good job of killing their skating."
New York (42-21-3) moved back into first place in the Metropolitan Division, two points ahead of the New York Rangers, who have two games in hand. Halak made 28 saves. The Islanders won for the 19th time on the road and ended a two-game winless streak.
"You've just got to stick with it," Islanders captain John Tavares said. "It happens. There's times the team is going to make plays, get some bounces or maybe you're not sharp for a couple shifts, so we just stuck with it. I think for the most part after that early shaky start, we were the better team and we created opportunities."
Nashville (41-18-7) remained four points ahead of the St. Louis Blues in the race for first place in the Central Division despite losing its fifth game in a row and third straight at Bridgestone Arena.
"I really think we've got to look at the team," Predators coach Peter Laviolette said. "We'll get out of this together. Our guys have done an excellent job of staying tight as a group. There's lots of things that could have been better tonight as well, still. We still gave up scoring chances, and we gave up chances point-blank in front, so there's things that we can do better. I think it's important that we stick together right now."
Tavares gave the Islanders a 1-0 lead at 11:24 of the first period on a wraparound goal. Tavares got the puck behind the net, came out to Rinne's left and banked a shot off the goaltender and into the net for his 31st goal, matching his career-high. It was the 10th consecutive game in which the Predators surrendered the first goal of the game.
Rookie Anders Lee gave New York a 2-0 lead at 18:30 when he redirected defenseman Nick Leddy's power-play shot past Rinne for his 22nd goal. Tavares set up Leddy at the right point, and Lee changed the direction of the shot and got it past Rinne.
"He's [got] net presence," Islanders coach Jack Capuano said of Lee. "You want to score goals, you want to learn how to score goals, you go right there. That's where you score. He plays well defensively and plays our structure, but the one thing that I really like about him is that he plays with [Josh Bailey] and [Tavares] and they know exactly where he's going to be all the time. He goes to those dirty areas, he's a big guy and he's tough to move."
Colin Wilson scored with 1.2 seconds remaining in the period to make it 2-1. The Islanders failed to clear their zone; Wilson found the puck in a scrum in front of the net and slapped it through traffic and into the net for his 20th goal of the season.
Leddy gave the Islanders a 3-1 lead at 8:39 of the third period with a wrist shot from the left circle. His shot trickled under Rinne's arm and into the net for his ninth goal of the season.
"I think that's one of his strong suits," Nelson said of Leddy. "I think he brings that to the table every day. He's such a great skater that he makes it look effortless and he jumps up when he sees an opportunity. He's got a quick shot, a good shot. He's been huge for us all year. Power play, five-on-five, he's one of the guys we lean on to chip in offensively and also shut guys down."
Ribeiro got the Predators within a goal at 10:16 on a redirection that got past Halak. Defenseman Ryan Ellis took a shot from the right point that hit Ribeiro in the forearm and deflected into the net for his 13th goal of the season. The goal stood after a video review.