Fleury stopped Brad Richards and Vincent Lecavalier to start the shootout, but St. Louis skated down the slot and faked out Fleury by going to his backhand and then slipping the puck between the goalie's legs.
Crosby had a chance to send the shootout to another round but, like Eric Christensen and Evgeni Malkin before him, couldn't get the puck past Johan Holmqvist. Crosby's hard wrist shot went wide of the net.
Holmqvist has stopped all 10 skaters he has faced in shootouts this season.
Nick Tarnasky put Tampa Bay ahead 2-1 with about six minutes left in the third period by lifting a wrist shot from the high slot past Fleury for only his second goal. Mark Recchi - who also set up Crosby's goal - tied it on a power-play goal set up by Crosby with 2½ minutes left in regulation.
The Penguins, their three-game winning streak stopped, did almost nothing offensively against the team that has allowed the fewest shots in the league until Crosby's goal late in the second period tied it 1-1.
Nolan Pratt's shot on a good scoring chance slammed off the rear boards, allowing Recchi to start a break the other way. With defenceman Cory Sarich trying to defend on the 2-on-1, Recchi came down the right wing boards and threw a pass toward Crosby in the left circle.
Crosby slipped to one knee as he stretched out, then two knees and, finally, was on his stomach as the puck arrived. Crosby not only got his stick on it but was able to lift it over Holmqvist and into the side of the net with five seconds remaining for his 20th and most acrobatic goal of the season.
The sellout crowd, the Penguins' sixth in a row and 10th in 12 games, gave Crosby such a lengthy standing ovation that it caused a brief delay in the ensuing faceoff.
Crosby's goal marked the second time in as many games the Penguins scored in the final five seconds of the second period. Malkin scored with three seconds remaining in a 4-2 victory in Buffalo on Friday.
Tampa Bay, going 2-2 on a circuitous Montreal-Minnesota-Colorado-Pittsburgh road trip, had taken a 1-0 lead on Ryan Craig's power-play goal about seven minutes into the second period. Craig also set up Tarnasky's goal.
Pittsburgh tied it late in the third when Crosby's pass from the right circle found Recchi at the top of the crease, and the puck deflected off Recchi and Sarich's skate and into the net. Recchi's 10th goal and 1,300th career point was initially waved off, but a lengthy video replay determined Recchi didn't kick the puck into the net.
Tampa Bay has won 10 in a row against Pittsburgh dating to 2003, including an overtime 4-3 decision in Mellon Arena on Lecavalier's goal on Nov. 8. The Lightning won this one despite failing to score on a full-length power play in overtime.