Florida, which has lost three straight (0-2-1), was 0-for-4 on the power play and allowed two shorthanded goals.
"Give their goalie credit. He played really well, ours didn't," Panthers coach Bob Boughner said. "Those guys, they've been good all year on the power play. Tonight, when we needed them the most, we're in a playoff push, you have two or three guys that are just out to lunch."
Pearson put the Penguins ahead 1-0 at 1:29 of the first period on a wrist shot from the right face-off circle before Rust made it 2-0 on Pittsburgh's third shot of the game, at 3:03. Jake Guentzel chipped the puck to himself through Aaron Ekblad's legs at the blue line before passing to Rust, who scored on his backhand.
Rust's shorthanded goal off a wrist shot made it 3-0 at 3:04 of the second period. It was his 11th goal of the season.
Sheahan extended the lead to 4-0 with a shorthanded goal at 8:56.