Ilya Bryzgalov made 21 saves, shutting down the NHL's highest-scoring team to help the Flyers win their second straight game and improve to 4-6.
The Lightning, who entered averaging 4.9 goals per game, had dominated the series, winning seven of the previous nine meetings.
Benoit Pouliot scored for Tampa Bay. Steven Stamkos was held without a point for the first time this season as the Lightning (6-3) lost their second consecutive game.
Less than 2 minutes after Pouliot tied it in the third period, Sestito scored the go-ahead goal on a nearly identical move.
Sestito skated down the center on a break, took a pass from Ruslan Fedotenko and slipped a backhander under Andres Lindback's pads for a 2-1 lead with 8:49 left.
Sestito gave the Flyers a 1-0 lead 4:25 into the second period. He took a crossing pass from Jake Voracek, skated in from the right circle and niftily went to his backhand to beat Lindback.
Before that, Sestito's last NHL goal was on Dec. 23, 2010, for Columbus against Vancouver.
Bryzgalov was sent crashing to the ice after stopping Cory Conacher's shot in the third period. He got up slowly and earned a loud ovation from the crowd.
But shortly afterward, Pouliot lifted a shot over Bryzgalov's right shoulder to tie it.
Bryzgalov, who had an up-and-down first season in Philadelphia after signing a $51 million contract, has been excellent this year. He lowered his 2.40 goals-against average and improved his .920 save percentage.
Bryzgalov made an outstanding save on Stamkos, sliding post-to-post to keep the game scoreless late in the first.
Philadelphia's Zac Rinaldo decked B.J. Crombeen with a flurry of fists after they dropped their gloves a few minutes into the game. Crombeen seemed woozy as he slowly got up and left the ice. He didn't return as a precaution.
NOTES: Rob Zombie, the Hollywood director and rocker, was at the game working on ideas for his movie on the Broad Street Bullies. ... Flyers GM Paul Holmgren said there's been no change in Wayne Simmonds' condition and he remains day to day. Simmonds sustained head and neck injuries on a hit from Washington's John Erskine last Friday. ... The Lightning began a stretch of four road games in six days. ... Injured Flyers forward Scott Hartnell was on crutches in the press box. Hartnell is out with a broken left foot.