It was Cirelli's second hat trick in the NHL and first since Jan. 17, 2020.
Nikita Kucherov had a goal and two assists for the Lightning (47-22-6), who are 7-1-2 in their past 10. Jake Guentzel and Gage Concalves each had two assists, and Andrei Vasilevskiy made 21 saves.
Egor Chinakhov scored twice, and Rickard Rakell extended his goal streak to three games for the Penguins (38-22-16), who had won two straight and three of four. Samuel Girard and Kris Letang each had two assists, and Stuart Skinner made 27 saves.
Cirelli gave the Lightning a 1-0 lead at 5:06 of the first period. Concalves intercepted a pass from Penguins forward Ben Kindel in the defensive zone and put a snap shot off Skinner's blocker from above the right face-off circle. The puck rebounded to Cirelli in the slot, where he buried it to the glove side.
Rakell tied it 1-1 at 7:01. After Girard skated behind the net, the puck deflected off two Lightning defenders to Rakell, who went skate-to-stick behind his back and stuffed it past Vasilevskiy at the left post.
Chinakhov put Pittsburgh up 2-1 at 16:10 when he lifted a backhander from the right circle under Vasilevskiy's right arm.
Cirelli evened the game 2-2 just 11 seconds into the second period, scoring his second goal while on a short-handed breakaway with a backhand shot over Skinner's left pad.
Brayden Point gave Tampa Bay a 3-2 lead at 14:08. Guentzel followed the rebound of his shot behind the net and knocked it out in front, where Point chipped the puck over Skinner's glove.
Zemgus Girgensons increased it to 4-2 with his 100th NHL goal at 19:05. Yanni Gourde sent a backhanded pass from behind the net to Girgensons in front, where he went forehand-backhand and tucked the puck around Skinner's left pad.
Kucherov made it 5-2 on a sharp-angle wrist shot from the right circle at 13:50 of the third period.
Cirelli completed the hat trick with an empty-net goal at 15:53 to push it to 6-2.
Chinakhov cut the deficit to 6-3 while on the power play with his second goal at 19:58.
