The Rangers got goals from Artem Anisimov, Michael Del Zotto and Brandon Dubinsky. Henrik Ludnqvist gave up five goals on 31 shots.
The Rangers had taken a 4-3 lead with just 7:14 to play in the third period when Anisimov scored on a one-timer off a feed by Marian Gaborik.
But the Islanders capitalized on penalties to Ryan Callahan and Marc Staal to score two power-play goals for a 5-4 lead.
Parenteau got the tying goal when he banged in a rebound and less than a minute later Comeau's one-timer put the Islanders ahead.
Nielsen got the final tally on an empty-netter that sealed the win.
2014: The Year in Music [PHOTOS]