Vanek, obtained from the Buffalo Sabres in a trade last week, also assisted on a goal as the Islanders recorded a second straight win for the first time this season.
Andrew MacDonald and John Tavares each scored for New York.
Islanders goalie Kevin Poulin yielded one goal on 27 shots.
Dougie Hamilton scored and backup goaltender Chad Johnson, making his second start of the season, stopped 31 shots for the Bruins.
Celebrity Families of 2014 [PHOTOS]