Moulson scored 2 minutes apart in the first period and added his 12th of the year during a wild early second period, which saw a total of five goals scored within a 6-minute span.
But his fourth goal with 2:50 remaining in the middle frame was the game-winner, as he and P.A. Parenteau came in on net after a Sheldon Souray turnover at the blue line and Moulson slid home a centering pass, breaking a 4-4 tie.
Dylan Reese got the Islanders' other goal and John Tavares added three assists for New York.
Andrew Raycroft surrendered five goals on 27 shots in the loss.