Grabner had not scored since the opening game of the season, but he produced a shorthanded tally in the first period and clinched the win with an empty-net goal in the final minute.
The Rangers overcame an early 2-0 deficit on consecutive goals by Benoit Pouliot, Derek Dorsett and Derek Stepan. The last of those goals came midway through the second period.
Brian Strait tied it for the Islanders with 8:34 to go in the middle stanza and Thomas Vanek put them in front on the power play 1:18 into the third.
Evgeni Nabokov made 21 saves in helping bring the Islanders only their second win in 15 games.
Henrik Lundqvist allowed four goals on just 19 shots for the Rangers, who are involved in the longest homestand in the history of the franchise. The Rangers have gone just 1-4-2 during the nine-game stretch of home contests.