Neal scored the tying and go-ahead goals for the Penguins, who broke a two-game losing streak and improved to 3-3-0 at home.
Stephane Da Costa and Jim O'Brien scored for the Senators, who fell for the fifth time in seven games.
Craig Anderson made 26 saves for the Senators, who were rocked by the news Karlsson could possibly miss the rest of the season after being cut on his Achilles tendon by the skate of Pittsburgh's Matt Cooke.
The injury came in the second period when Karlsson -- the NHL's reigning Norris Trophy winner -- was pushed into the boards by Cooke, whose skate came off the ice and cut into the defenseman.
The Senators announced after the game Karlsson will have surgery and is out indefinitely.