The Rangers overcame a two-goal deficit to create overtime and both sides produced a score in the opening round of the shootout. Moulson put the Islanders back in front in the fourth round and Evgeni Nabokov clinched the win by stopping the Rangers' John Mitchell.
The Islanders, who had lost their previous two games, improved their record against the Rangers this season to 2-3. The teams will meet once more in March.
Moulson opened the scoring 6:07 into the contest with his 26th goal of the season and PA Parenteau tallied twice for the Islanders, who owned a 3-1 lead midway through the contest.
The Rangers then got goals from Marian Gaborik and Derek Stepan during the latter portion of the second period to tie it.
Nabokov was credited with 33 saves. Martin Biron stopped 24 shots for the division-leading Rangers.