Tampa Bay lost for the sixth time in its past seven games that caused the Lightning to hold a players-only meeting after the game.
Star forward Claude Giroux suffered a head injury in the second period and left the game.
Before the injury, Giroux had two assists to add to his NHL-high point total of 39 (16 goals and 23 assists) and extended his point streak to five games (three goals and seven assists for 10 points).
Giroux took an inadvertent knee to the head by Simmonds at center ice in the last minute of the second period and did not return for the final period. The Flyers are already without defenseman Chris Pronger with concussion symptoms.
"I didn't even know I hit him," Simmonds said. "It was a split-second thing. I was trailing (Giroux) by a couple of steps, and all of sudden, he went down right in front of me. It was definitely bad, and I wish it hadn't happened."
Flyers general manager Paul Holmgren said Giroux was held out of the third period for "precautionary reasons."
Holmgren said "This is precautionary and we want to err on the right side of caution."
Holmgren said Giroux's condition would be evaluated on Sunday, but said, "It wasn't really a headache" Giroux was experiencing.
"Claude has been playing on a different stratosphere for us and this is part of the game," Holmgren said.
Tampa Bay's Vincent Lecavalier scored a power-play goal to pull the Lightning within 3-2 in the opening minute of the third period.
But Philadelphia answered a few minutes later when Matt Carle had a power-play goal that made it 4-2.
Scott Hartnell had an empty-net goal to cap the scoring. It was the fifth straight game Hartnell scored a goal.
Trailing 1-0 after the first period, Briere tied the game on his ninth goal of the season at 2:38 into second period, beating Lightning goalie Mathieu Garon over his right shoulder.
Then Jagr added his 10th, beating Garon with a slap shot. The score was Jagr's 656th career goal, tying him for 11th in career goals with Brendan Shanahan.
Simmonds topped off the three-goal run with a power-play goal with slightly over five minutes left in the second. Simmonds received considerable help, when he backhanded a shot that ricocheted off the skate of Tampa Bay defenseman Brett Clark and bounced into the net over Garon.
Philadelphia outshot the Lightning 16-7 in the second period.
"I don't think we were poor in the first period, we just didn't have that same zip that we had in the past, and I was really happy to see us in the second period really playing our game," Flyers coach Peter Laviolette said. "I was really happy with the way we rebounded. We could have scored four, or five, or six goals there, without a few bounces not going our way and some great saves."
Garon made a nifty stick save in the last minute of the second. The Flyers' Sean Couturier jabbed at a rebound when Garon flicked the puck away inches from crossing the goal line.
"Turnovers led to two goals, you have to give it to them they played well, let's not undermine what they did, but we killed ourselves, sabotaged ourselves," Lightning coach Guy Boucher said.
"We had tons of scoring opportunities to score in the third and we just didn't shoot. If we don't give up the turnovers we don't give up the two goals it's a closer game, I don't say we don't win it, but it's a closer game."
Tampa Bay controlled much of the opening period. Ryan Shannon gave the Lightning a 1-0 lead on his second goal of the season halfway into the first period.
The play was made possible by the hustle of Dominic Moore, who beat the Flyers to the puck off the boards, then threaded a perfect pass through the legs of the Flyers' defense to find Shannon.
Flyers goalie Ilya Bryzgalov wasn't able to slide over fast enough, and there was Shannon all alone to poke the puck into a wide-open net.
With less than 13 minutes in the game, Laviolette pulled Bryzgalov in favor of Sergei Bobrovsky. Bryzgalov had made 14 saves. After the game, Holmgren said Bryzgalov suffered a lower-body injury.
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PHILADELPHIA, PA - DECEMBER 10: A view of the Philadelphia Flyers locker room is seen prior to their game against...