The Bruins grabbed a 1-0 lead 4:52 into the game after Thornton backhanded in the rebound off Doug Hamilton's shot.
Aucoin got to a misplaced clearing attempt in the high slot and unloaded a slap shot past Rask's glove to give the Islanders a 2-1 edge midway into the second, but the Bruins tied it less than 4 minutes later. Campbell controlled a rebound in front and the puck past Rick DiPietro.
Bergeron scored on a breakaway with 6:27 to play, shifting a few times before tucking in a forehand shot.
Chara hit the left post in the closing minute with a backhand shot from the blue line after the Islanders pulled DiPietro for an extra skater.
In Detroit, Henrik Zetterberg broke a tie midway through the second period, and Todd Bertuzzi scored his second goal of the game early in the third to lift the Detroit Red Wings to a 5-2 win over the Minnesota Wild.
Pavel Datsyuk, who assisted on Zetterberg's goal, put Detroit ahead by two late in the second period. Datsyuk also set up Bertuzzi on his second goal that gave the Red Wings a 5-2 lead.
In Anaheim, California, Cory Schneider made 30 saves in his fifth career shutout, Mason Raymond scored twice, and the Vancouver Canucks avenged last week's blowout loss to Anaheim with three power-play goals in a 5-0 win over the Ducks.
In other games Friday, Ilya Kovalchuk scored 4:39 into overtime, and the New Jersey Devils won their third straight game to start the lockout-shortened season, 3-2 over the winless Washington Capitals and Ryan Malone scored a pair of third-period goals as Tampa Bay beat Ottawa 6-4.