Financial terms were not announced.
The 26-year-old forward had 35 points (19 goals, 16 assists) in 82 games with the Islanders last season. He was a restricted free agent.
Nelson, who scored at least 20 goals in three straight seasons from 2014-17, has missed two games in the past four seasons.
Selected by New York in the first round (No. 30) of the 2010 NHL Draft, Nelson has 188 points (99 goals, 89 assists) in 398 regular-season NHL games and seven points (three goals, four assists) in 18 Stanley Cup Playoff games.