Semyon Varlamov made 31 saves and Josh Bailey had a goal and an assist for New York (20-7-2), which has outscored the Lightning 10-3 in two victories this season.
"A real good team win. We needed a big goal and we got it," Islanders coach Barry Trotz said. "The special teams were huge, good goaltending tonight. Our compete level through the lineup was very good against a very good hockey team.
"There's always challenges in this building. Special teams, you get a shorthanded goal, a power-play goal, just all those elements in your game that you need. At the end there, we had our killer instinct."
WATCH: [All Islanders vs. Lightning highlights]
Steven Stamkos scored for Tampa Bay (14-11-3), which lost twice on a three-game homestand and is 1-4-0 in its past five home games. Curtis McElhinney made 19 saves.
"It's tough because we're working hard and putting a great effort forward," Lightning defenseman Kevin Shattenkirk said. "Obviously we're not getting the results we want, but we can't change anything because it's a blueprint for success."
Nelson gave the Islanders a 3-1 lead with an unassisted goal at 7:31 of the third period, then made it 4-1 with a wrist shot from the slot at 14:41.