John Tavares scored in regulation and Jaroslav Halak had 33 saves through overtime and then stopped all three Rangers attempts in the shootout to give the Islanders their fourth win in five games (4-0-1).
Viktor Stalberg scored the tying goal in the second period, and Henrik Lundqvist had 36 saves for the Rangers, who lost for the fourth time in five games (1-3-1).
Tavares gave the Islanders the lead with a 5-on-3 power-play with 7:38 to go in the second period.
Stalberg redirected Marc Staal's shot past Halak to tie it with 1:23 remaining in the period.
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