Laughton started Tuesday on the third line with van Riemsdyk and Konecny but earned a chance on the top line with Giroux and Voracek after Couturier left the game.
"I've always said Scotty is one of our character players," Vigneault said. "He cares, he wants to do well. He especially wants the team to do well. When [Couturier] wasn't able to come back, I looked at our options and how certain guys were playing and I thought he'd be the best fit with [Giroux and Voracek]."
Trailing 3-1 late in the third period, the Islanders rallied with two goals in 1:33 to force overtime.
Nelson made it 3-2 with a one-timer with 4:14 remaining in the period, and Brassard tied it 3-3 with 2:41 left.
"We just started to get chances," Islanders forward Jordan Eberle said. "We got pucks in, we started to play the way we know we can play. We started to get a lot of chances. [Bailey] made a great play to [Nelson] and we found one and then we found the one from [Brassard], who's been stepping up lately. So that's just the character in our team. There's no quit in us."
The Islanders took a 1-0 lead at 1:20 of the second period on a power-play goal by Barzal, who had a shot by Bailey bounce off him and past Hart. The goal was upheld after Vigneault challenged for goaltender interference.
Bailey has 15 points (two goals, 13 assists) this postseason, the most by a New York player since Ray Ferraro (20) and Steve Thomas (17) in 1993.
Giroux tied it 1-1 at 15:45 of the second when he deflected a point shot by Philippe Myers between Varlamov's pads.
Van Riemsdyk put Philadelphia ahead 2-1 with 1:42 remaining in the second when he finished a 3-on-1 rush.