Mantha gave Pittsburgh a 1-0 lead at 12:09 of the first period. Brazeau passed from below the goal line to Mantha, who beat Sorokin blocker side from in front for his 20th goal of the season and sixth in his past six games.
Horvat tied it 1-1 at 18:41, batting in a loose puck at the right side of the net while falling to the ice.
Just 1:15 later, Schaefer sent a slap shot from the point past Skinner’s glove to give the Islanders a 2-1 lead at 19:56.
Chinakhov made it 2-2 at 3:52 of the second period, scoring from the left hash marks after Tommy Novak found him open from behind the net.
Sorokin got his left pad on a Connor Dewar breakaway try at 6:18 after the Penguins killed off an Erik Karlsson delay-of-game penalty.
Rust then put the Penguins in front 3-2 at 14:09. He took a pass from Crosby behind the net and banked a shot off Sorokin at the right post from below the goal line.
Barzal’s one-timer from inside the blue line beat Skinner glove side through a screen to tie the game 3-3 at 8:37 of the third.
“What a ride that was,” Barzal said. “I thought we came out, had a great start, and then we were just always down a goal, and then we get it tied up late, and they get one in front of the net there, and we kind of have a bad power play. And tonight was big, just for the morale in this room, just to know that it doesn't really matter how the game goes -- up [and] down the entire night. I think we knew the whole way we were getting it done. We felt super confident every period.”