Sillinger, who got the only assist on Jeff Tambellini's first goal of the season in the opening period, shook off an injury and scored his 19th at 10:56. That gave the Islanders at least one standings point for the 11th time in 12 games (7-1-4).
New York, starting its second seven-game homestand of the season, moved into a three-way tie for eighth place in the Eastern Conference - one point behind Carolina.
Boston is mired in 12th place, eight points behind the playoff cutoff. This was not how the Bruins wanted to begin a season-high, six-game road trip.
Rick DiPietro made 24 saves and allowed two goals or fewer for the sixth time in seven games. Randy Robitaille sealed the win with a goal with 3:15 left and another into an empty-net.
Glen Murray scored for Boston, which lost for just the second time in six games. Tim Thomas finished with 34 saves.