Marc-Andre Fleury, who earned his seventh consecutive win when starting against Tampa Bay, made nine of his 28 saves in the third period, and rookie defenseman Brian Dumoulin scored his first NHL goal by placing a snap shot through Lightning goalie Evgeni Nabokov's five-hole 7:47 into the third to put the Penguins (20-6-4) up three goals.
Rookie forward Bryan Rust also scored his first NHL goal when he placed a wrist shot over Nabokov's glove to give Pittsburgh a 2-1 lead 2:55 into the second period.
"It was a tough battle tonight," Fleury said. "They're a good team, and it was nice to see those two guys get their first goals in the NHL. They were key for our win."
Rust said he and Dumoulin joked with each other before the game about each planning to score his first goal.
"It's obviously something I've been looking forward to for a long time now," Rust said. "I was just happy to get into a game or two and I was just in the right place at the right time. [Andrew Ebbett] found me in the slot, and I was just able to fire it home."
Nikita Kucherov scored his 12th goal, second on Tampa Bay (19-10-3) behind Steven Stamkos' 17, with 3:02 left to pull the Lightning within two goals. Stamkos did not register a shot until 2:06 remained in the third.
"Our line didn't generate enough," Stamkos said. "I don't think it was [the Penguins] in particular. I think it was more us. When you're not creating chances, you're obviously not getting shots on net. So, it's a little frustrating."
Pittsburgh played without center Sidney Crosby, who was diagnosed with the mumps over the weekend, for the third consecutive game. The Penguins are 92-47-18 in 157 games without Crosby since his NHL debut in October 2005.
The Penguins used two second-period goals to carry a two-goal lead into the intermission after surviving a first period dominated by the Lightning. After Rust's goal, Steve Downie added to the Penguins' lead with 33.3 seconds left in the second.
A shot from forward Patric Hornqvist jumped off of Nabokov's chest and over his shoulder before bouncing off Lightning defenseman Matthew Carle's helmet and landing at Downie's skates. Downie backhanded the rebound into the open net for Pittsburgh's first power-play goal in four games.
"I thought we carried the play for a lot of the game, to be honest," Lightning coach Jon Cooper said. "We scored three goals on ourselves. They scored four; three went off us in the net. It's hard to win when you put three in the net. We put three in our net and two in theirs. So, maybe we won 5-1."
Nabokov replaced goalie Ben Bishop, who sustained a lower-body injury, at the start of the second.
Cooper said it's not likely Bishop will play Tuesday against the Philadelphia Flyers.
"We play in less than 24 hours. He pulled himself out tonight. It'd be hard to go in less than 24 hours," Cooper said, according to the Lightning website.
Bishop allowed a shorthanded goal by Penguins forward Brandon Sutter with 51 seconds remaining in the first period. He made four saves on five shots before leaving the game, which was tied 1-1 at the end of the first.
Tampa Bay controlled the first, but a few key saves from Fleury kept the Penguins from falling behind.
Fleury made two impressive stops on forward Alex Killorn, the first of which came 3:43 into the period. Killorn collected a rebound and backhanded a shot with Fleury out of position, but the Penguins goalie dove to make the save under his left armpit.
Killorn was denied again when Fleury stopped a second backhand attempt with 4:47 remaining in the first.
"It's pretty frustrating," Killorn said. "They trickled almost by him. … It's pretty frustrating. I don't get too frustrated after that because I'm happy I get those chances, but I need to score one of those, team-wise, to get us going."
Pittsburgh took advantage of Fleury's effort by taking a 1-0 lead on Sutter's second shorthanded goal of the season. The puck dribbled away from Lightning forward Valtteri Filppula in the corner behind the Penguins net and to Pittsburgh defenseman Rob Scuderi, who sent Sutter streaking down the ice with a pass.
Sutter took a wrist shot that clanged off the left post and crossbar before bouncing into the back of the Tampa Bay net.
The Penguins' lead lasted 25 seconds. Rookie forward Jonathan Drouin tied the game with the power play still in effect. Fleury lost sight of the puck after forward Tyler Johnson sent a shot into the crease. Drouin shoveled the rebound behind Fleury with 26.2 left in the first.
The Penguins struggled to generate any offense early and didn't register a shot until Blake Comeau's snap shot with 9:02 left in the first period. The Lighting outshot the Penguins 12-5 in the first after impressively killing two Pittsburgh power plays.
Penguins coach Mike Johnston said another unlikely hero, forward Bobby Farnham, playing in his second career game, was a catalyst in Pittsburgh pulling through the first.
"Early in the game, when we didn't have a lot going on, he was the one bright light," Johnston said. "He came out on the ice, got in on the forecheck and stirred things up. He can do that. And as you guys know, there are a lot of games. The schedule's fairly heavy, having a guy like that in your lineup that just picks up your team, picks up the crowd, picks up the energy in the building. … He just plays the same way every night."