Neal's second goal, his 17th of the year, midway through the second period erased a 3-2 deficit and Dupuis found the net 7 minutes later for the go-ahead tally.
Marc-Andre Fluery turned aside 24-of-27 shots to earn the victory.
Kyle Okposo notched a goal and an assist, while Milan Jurcina and David Ullstrom also scored for the Islanders.
Al Montoya allowed six goals on 31 shots for New York.
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