Carter Hart made 49 saves, and Kevin Hayes had a goal and an assist for Philadelphia, the No. 1 seed in the East. Each of the Flyers' three wins in the series has come in overtime.
"We've had two other games where we've won in overtime, so that was the message in locker room: Don't change anything, just play our game," Hart said. "That's what we did. We just needed to funnel pucks to the net, crash hard, and that's how you score goals, just grinding it out. That's what it was there at the end."
WATCH: [All Flyers vs. Islanders highlights]
Game 7 is in Toronto, the East hub city, on Saturday (7:30 p.m. ET; NBC, CBC, TVAS).
Teams that win Games 5 and 6 after trailing a series 3-1 are 29-28 (50.8 percent) winning a best-of-7 Stanley Cup Playoff series.
Provorov, who led the Flyers with 38:15 of ice time, won it with a wrist shot from beyond the circles after a pass from Hayes.
"There's no doubt that this wasn't one of our better games," Flyers coach Alain Vigneault said. "But at the end of the day we found a way to get the job done, we found a way to come back in the third and tie up the game against a team that doesn't give up a lot of leads. [It] took us two overtimes, but we found a way to make a play in the second overtime. Got a great shot and a goal that permits us to have a chance.
"That's what we wanted. We wanted to have a chance. If someone had asked me at the beginning of this series against such a strong opponent, one game, winner take all, I would have taken that."