Montoya turned aside all 35 shots he saw in registering his first shutout since the 2008-09 season when he was with Phoenix. Making only his fifth career NHL start, Montoya was especially busy in the second period when he turned aside 17 shots.
Moulson's goal less than 7 minutes into the second period gave the Islanders a 2-0 lead and he later added an insurance goal, his 23rd of the season, 13:40 into the final frame.
Frans Nielsen also scored as New York won for the fifth time in six games.
Jonathan Quick stopped 20-of-23 shots for the Kings, who are battling for one the last playoff spots in the Western Conference.