The Bolts led for most of the opening period after Stamkos scored his fourth goal in three games, but Sidney Crosby equalized on an odd man rush just before intermission.
Danton Heinen put Pittsburgh in front for good with a power play goal early in the second before two goals 1:22 apart from Jake Guentzel and Jeff Carter in the opening stages of the third extended the deficit to three.
Bryan Rust and Brayden Point traded late goals before Rickard Rakell found the empty net to round out the scoring.
Brian Elliott made his first start of the season and was sharp despite the loss, finishing with 39 saves.
The loss drops Tampa Bay to 1-2-0 on the season and forces them to settle for a split in this back-to-back against the Blue Jackets and Penguins.
The Lightning now return to Tampa for their home opener on Tuesday night against the Flyers.
Scoring Summary
1st Period
TBL 1, PIT 0
4:10 Steven Stamkos (4) - Victor Hedman, Nikita Kucherov - PPG
The Lightning captain continued his hot start with a game-opening power play goal. It was vintage Stamkos from his office, with Victor Hedman dishing the puck to him atop the left circle, and Stamkos uncorking a massive one-timer that beat Tristan Jarry.
TBL 1, PIT 1
17:00 Sidney Crosby (2) - Jake Guentzel, Brian Dumoulin
On a 2-on-1 rush, Jake Guentzel played a beautiful saucer pass over a sliding Erik Cernak to Sidney Crosby on the left. Crosby cut across the face of goal and tucked a backhander by Brian Elliott's outstretched arm.
2nd Period
PIT 2, TBL 1
4:11 Danton Heinen (1) - Jason Zucker - PPG
In the dying seconds of Pittsburgh's second power play of the night, a bouncing puck kicked to Danton Heinen at the bottom of the right circle. From a sharp angle, Heinen slipped a shot past Elliott to give the Penguins their first lead of the night.
3rd Period
PIT 3, TBL 1
2:28 Jake Guentzel (2) - Kris Letang, Sidney Crosby
The Lightning lost a defensive zone draw and Kris Letang's shot from the point was tipped through Elliott's five-hole by Guentzel.
PIT 4, TBL 1
3:50 Jeff Carter (1) - Kasperi Kapanen, Pierre-Oliver Jospeh
Kasperi Kapanen chased down a puck deep in the Lightning zone and dished back to the middle for a trailing Jeff Carter, who scored to open a three-goal Pittsburgh lead.
PIT 5, TBL 1
13:31 Bryan Rust (2) - Sidney Crosby Evgeni Malkin - PPG
Bryan Rust scored Pittsburgh's fifth-consecutive goal with a tip-in at the side of the net. With the Lightning having to kill off a Brandon Hagel holding penalty, Crosby's initial shot went in off Rust stationed to Elliott's right.
PIT 5, TBL 2
14:03 Brayden Point (1) - Nikita Kucherov, Steven Stamkos
Facing a big deficit, the Lightning answered 32 seconds after Rusts' goal with Brayden Point registering his first of the season off a Nikita Kucherov feed.
PIT 6, TBL 2
15:55 Rickard Rakell (1) - Unassisted - ENG
Rickard Rakell reestablished the Penguins' four-goal lead with an empty netter after Jon Cooper pulled Elliott in favor of the extra attacker.
