Skinner fired a shot past Evgeni Nabokov's blocker for the deciding tally after Riley Nash had also found the net in the tiebreaker.
The Islanders, however, secured a standings point when Kyle Okposo scored with 1:01 left in regulation to forge a 3-3 tie.
The result, coupled with Winnipeg's regulation loss to Washington Tuesday, put New York into the post-season for the first time since 2007.
The Islanders remained without a regulation loss in their last 11 games, going 8-0-3 over the span.
Celebrity Families of 2014 [PHOTOS]