The first one came just 1:16 in. Crosby took a pass from Kunitz, skated alone down the left side and roofed a short-side wrist shot over Lindback's shoulder from the faceoff dot.
Crosby's second goal came 6 minutes later. Lindback had trouble trying to glove Martin's shot from the point. Crosby got a stick on the puck, which bounced between Lindback's legs and into the net after the Lightning goalie kicked it in.
Martin scored with 33 seconds left in the first period, tapping the rebound from Cooke's initial shot into a partially empty net to give the Penguins a 3-0 lead.
The Lightning, after a dismal opening period, needed just 3:14 to score twice in the second.
Brewer got the first one 1:37 into the period on a tip-in from Stamkos, who followed it up by scoring the second — his fifth straight game with a goal — just 20 seconds into a two-man advantage.
Stamkos, with Pittsburgh's Simon Despres and Craig Adams in the penalty box, wasted little time scoring on the power play, ripping a one-timer from the left side past Fleury to make it 3-2.
"They got right back into it and our three-goal lead evaporated," Bylsma said. "I like how we stabilized the ship."
The Penguins regained a two-goal lead, scoring on their own two-man advantage. It was the 11th straight game in which they scored a power-play goal, and the first time in five games Tampa Bay allowed a goal with the man advantage.
Kris Letang rushed to keep the puck in the zone at the blue line, sending a spinning pass to Crosby, who set up Bennett's first NHL goal, a one-timer past Garon.
"Beau was in a good spot there," Crosby said. "He's a pretty gifted player, he has a lot of talent and he found the open lane."
Conacher scored 3:15 into the third period, one-touching Hedman's pass past Fleury to make it 4-3. But the Lightning, whose 32 third-period goals rank No. 1 in the NHL, couldn't forge a tie despite Hedman's near miss in the final minute.
"They're a good team that has great offense and it's going to bite you if you give them the puck, and that's what happened tonight," Boucher said.
NOTES: Crosby moved into third all-time in team history with 637 points, behind Mario Lemieux and Jaromir Jagr. ... Cooke got his 200th career assist on Martin's goal. ... Both teams play again Tuesday. Tampa Bay returns home to host Buffalo, while Pittsburgh opens a three-game road trip at Florida. In an odd scheduling quirk, the Penguins' next home game is March 4 against Tampa Bay.