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.
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 scored four goals in a 6:17 span to take the lead.
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 Pittsburgh to within 3-2.
Ryan Shea tied it 3-3 at 13:29. Mantha sent a backhanded pass from below the goal line to Shea in the left circle, where he put a one-timer into the top right corner off the post and in.
Mantha gave the Penguins a 4-3 lead just 1:06 later at 14:35. Brazeau chipped the puck to Mantha, who got behind the defense for a short breakaway and slid a backhander through the five-hole.
Mantha then scored his second of the period at 17:18 to make it 5-3. He collected a cross-ice pass from Brazeau in the left circle, skated to the net and lifted a backhand shot past Sorokin's glove.
Avery Hayes extended it to 6-3 at 6:06 of the third period. Ben Kindel settled his own rebound and threw it out in front from behind the net and Hayes scored with Sorokin out of position.
Rakell made it 7-2 with his second goal at 7:54 when a wrist shot from Brazeau deflected in off his stick as he battled on his way to the net and chased Sorokin from the game.
Rust finished a snap shot from the slot for the 8-3 final at 16:16. Crosby recorded the 1,100th assist of his career on the play, becoming the eighth player in NHL history to reach the milestone.
