Sidney Crosby, Jake Guentzel and Danton Heinen each had a goal and an assist for the Penguins (34-14-8), who have won three straight. Tristan Jarry made 20 saves.
"It was one of our more complete efforts all year long," Pittsburgh coach Mike Sullivan said. "I thought we had contributions from everybody on both sides of the net. For me what was probably the most encouraging part was our third period, I thought we continued to play the game on our toes. We created a lot of offense through our defense."
Pierre-Edouard Bellemare scored for the Lightning (35-12-6), whose five-game winning streak ended. Andrei Vasilevskiy made 35 saves.
"This one is on us as a staff and on the players as well, it's unacceptable to come out like we did," Tampa Bay coach Jon Cooper said. "But Pittsburgh was exceptional and give them credit. They came into a game I'm sure they had circled on the calendar, and they imposed their will on us and we had no answer."
Heinen gave Pittsburgh a 1-0 lead at 3:46 of the first period when he scored on a loose puck in front of the net. Crosby extended the lead to 2-0 at 6:27 with a power-play goal, a wrist shot from the left circle.
"We knew we hadn't been great off the start and that's something that we focused on," Penguins forward Bryan Rust said, "If we play the way we can, the way we want to and the way that we know that we are capable of, I think we can do some pretty good things."