Jakub Voracek had two goals and Wayne Simmonds also scored for the Flyers, who were 7-1 in their previous eight games. The Flyers hadn't lost at home since Nov. 5 against Carolina.
The Lightning went 0-5-2 on the road between Nov. 16 and Dec. 14. But they're 7-1 since.
"When we lost seven in a row, we had some adversity, but we had some guys injured and those were tough buildings to play in," St. Louis said.
Goaltender Steve Mason had his worst game for Philadelphia, allowing five goals for the first time in 41 games with the team since being acquired last April. He was pulled for Ray Emery after the second period.
"I wasn't very good," Mason said.
It wasn't entirely his fault.
"We left Mase out to dry," captain Claude Giroux said, alluding to several turnovers that resulted into three goals on odd-man rushes.
After a quiet first period in which Killorn scored the only goal, the teams combined for seven in a defenseless, wide-open second.
Just 66 seconds after Voracek's second goal tied it at 3, Filppula scored his 18th goal on backhander off a rebound to give the Lightning the lead for good.
St. Louis fired a shot past Mason for his 20th goal with 57 seconds left in the period to give Tampa its third, two-goal lead. They held onto this one.
"You try to prepare the best you can to help the team win and to score 20 goals is something I'm proud of," said St. Louis, who has accomplished that 10 times in his career.
Tampa defenseman Matt Carle was a plus-6 with two assists and didn't even take a shot.
"He was outstanding," Lightning coach Jon Cooper said.
Down 3-1, the Flyers rallied when Simmonds scored a power-play goal on a rebound off Giroux's slap shot.
Voracek tied it a few minutes later on a nifty goal. He took a pass from Andrej Meszaros, skated down the right side, weaved through three defenders and beat goalie Anders Lindback with a shot past his left skate.
Killorn gave the Lightning a 1-0 lead less than five minutes in when he stole a pass near Philadelphia's blue line, skated around defenseman Mark Streit and wristed a shot over Mason's shoulder.
Philadelphia missed two chances in the first period when Tye McGinn hit the crossbar Voracek hit the post during a two-man advantage.
Tampa scored a pair of goals 2:07 apart early in the second sandwiched around a goal by the Flyers.
Purcell started the scoring outburst with a shot from the left side that went between Mason's legs. Then Voracek scored 83 seconds later on a 1-on-none breakaway after Giroux intercepted a pass and flipped him the puck.
But Malone made it 3-1 on a 2-on-1 breakaway off a pass by Nate Thompson. Malone added an empty-net goal in the third.
The Lightning's recent road success includes wins at Detroit, Vancouver, Calgary and Winnipeg.
"We have a good thing going on the road lately and we hope to keep it going," defenseman Andrej Sustr.
Lightning G Ben Bishop missed his third straight game with a sprained right wrist. He could return Monday at Columbus. ... The Flyers were seeking their first 11-game home winning streak since 2003. ... Mason was yanked from a game for the third time this season. ... Lightning improved to 6-7-1 vs. teams in the Metropolitan Division.
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PHILADELPHIA, PA - JANUARY 11: Alex Killorn #17 of the Tampa Bay Lightning celebrates his first period goal...