Bryan Rust had a goal and an assist, and Justin Brazeau had three assists for the Penguins (37-21-16), who have won two of their past three. Sidney Crosby, Kris Letang and Parker Wotherspoon each had two assists, and Mantha also had an assist. Arturs Silovs made 20 saves.
Pittsburgh moved one point ahead of New York for second place in the Metropolitan Division.
Mathew Barzal had a goal and an assist, and Adam Boqvist and Calum Ritchie each had two assists for the Islanders (42-28-5), who had won three of four and finished 3-2-0 on a five-game homestand.
Ilya Sorokin allowed seven goals on 29 shots before being replaced at 7:54 of the third period by David Rittich, who made two saves.
“We were just awful defensively,” New York defenseman Ryan Pulock said. “[We allowed] guys behind us, breakaways, losing puck battles ... halfway through that period, we were just awful defensively.”
Anders Lee gave the Islanders a 1-0 lead with a power-play goal at 1:27 of the second period. Boqvist's wrist shot from the blue line bounced off the end boards to Lee, who backhanded the puck past the left pad of Silovs at the right post.
Barzal made it 2-0 at 2:56, skating into the offensive zone on a 2-on-1 and burying a wrist shot from the right face-off circle to the far side for his first goal in 10 games.
Elmer Soderblom cut the deficit to 2-1 at 6:41 when he lifted the puck over a sprawling Sorokin from in front during a goal-mouth scramble.
Brayden Schenn pushed the lead to 3-1 at 9:17. Barzal put a shot on net from the left circle, and Schenn chipped the rebound over Silovs' glove as he glided through the slot.
The Penguins then scored four goals in a 6:17 span to take control of the game.
“The start of the second period, we got away from the things that I liked in the first a little bit,” Pittsburgh coach Dan Muse said. “Just some of the chances that we gave, just time and space, numbers. We've been in situations where we’ve been down before, and there's a lot of belief in this group that we can kind of settle things down and put together some shifts where momentum can start to come our way. You never know where it's going to come from. ... Honestly, tonight, it was a lot of pucks behind them. We were spending more time [in their zone]. We were setting ourselves up for a good chance to be in the offensive zone. I think that's when we were playing our best tonight.”
Rakell started the comeback with a short-handed goal at 11:01, deflecting Rust's pass through Sorokin's pads as he cut to the net to bring the Penguins to within 3-2.
“That was definitely a turning point in that game,” New York coach Patrick Roy said. “But after that, we had a chance to get back in the game, and I think we didn't match the game as well as we did in the first half of the game. It could be [due to] line changes, or it could be losing battles in the corner. Our sticks were not as good. It’s things that we've been doing so well lately, and I guess we just had a bad one in the system.”