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.
Chipotle plans first price increase in 3 years
Putin thinks Obama would save him if he were drowning