Don Sweeney scored again at 14:46.
Before Don Sweeney registered the winning goal at 14 minutes 39 seconds of the third period, the Rangers had a chance to go ahead.
Boston's Don Sweeney was second with 184.
His game-winner came when he dug out a rebound after a shot by Don Sweeney directly in front of Brodeur.
Sutter said, "There was no doubt in Don Sweeney's mind that he wanted to play, and so we made the right move."
Greenwald appreciated Don Sweeney's horn arrangement, which he called "joyous".
Don Sweeney, the usually reliable defenseman of the Bruins, was the guilty party on the third goal for the Islanders.
Leetch took a shot that caromed off Don Sweeney and right to Messier in the slot.
Randy Burridge and Don Sweeney also scored in the first period.
Don Sweeney, a defenseman, then scored with a wrist shot from the slot during a power play at 13:49.