Steven Stamkos scored twice, and Nikita Kucherov and Victor Hedman each had three assists for the Lightning (43-27-7), who had won five of six and 10 of 12 (10-1-1). Andrei Vasilevskiy made 23 saves.
Tampa Bay, which clinched a playoff berth on Friday, holds the first wild card from the East and fell four points behind the Toronto Maple Leafs for third in the Atlantic Division.
"These are tough games to play," Lightning coach Jon Cooper said. "I can sit here after the fact and lament how we played the first two periods, but these guys have pushed so hard for so long now. We're playing a team that has put themselves right in it. The League's too tight to not be on the same emotional wavelength. And we weren't."
Kucherov, who has 136 points (43 goals, 93 assists), moved five points ahead of Nathan MacKinnon of the Colorado Avalanche for the NHL scoring lead.
Nicholas Paul scored for the Lightning on a snap shot from the left circle 25 seconds into the third period to cut it to 4-2.
Anthony Duclair made it 4-3 at 7:23 with a shot blocker side from the high slot.
"I thought we just stayed with it," Crosby said. "It didn't feel like we did a lot wrong, but that's a team that if you give them time and space, they're going to convert."
Stamkos then tied it 4-4 at 10:45 with a one-timer on a power play, his second goal of the game.
"It's just disappointing because if we play like that all game, then we, maybe, have a different result," Stamkos said. "A case of too little, too late."
Crosby put Pittsburgh ahead 1-0 at 4:20 of the first period. Bryan Rust held the puck on a 2-on-1 rush, allowing Tampa Bay defenseman Darren Raddysh to slide by before passing to Crosby for a backhand tucked around the left post.
It was Crosby's 40th goal, the third time in 19 NHL seasons he's reached the mark.