Claude Giroux scored in the first period, and Sean Couturier's goal at 13:06 of the third tied it up at 2-2 to help the Flyers reach the shootout.
Steve Mason made 34 saves and stopped Craig Smith, Matt Hendricks and Matt Cullen in the shootout for the win.
Mike Fisher and Colin Wilson each scored for Nashville, which has lost two straight.
The Predators were playing their fourth game in six nights and without captain Shea Weber who is day to day with an eye injury after taking a puck to the face in Thursday night's 3-0 loss to the Oilers.
Their most experienced defenseman on the ice was Roman Josi with 116 career NHL games.
Mason gave the Flyers a chance to win in the shootout. He had a big save inside the final 30 seconds of regulation when Seth Jones passed to David Legwand for a shot in front.
In overtime, Wilson hit a post at 2:30, and Mason stopped him on a breakaway with about 30 seconds left. Matt Read was wide on a shot on Nashville goalie Marek Mazanec during the extra period.
In the shootout, Lecavalier was the first shooter for the Flyers and beat Mazanec with a forehand. Giroux was stopped at the post, but the Flyers didn't need a third shooter when Cullen missed.
Not only did Nashville not have their leading goal-scorer in Weber, but defenseman Kevin Klein also was out for the Predators with a lower body injury, their team leader with 58 blocked shots.
The Flyers missed a prime chance at scoring on their lone power play of the first period when a shot slipped under Mazanec and was lying just near the line only to be poked away by Philadelphia forward Jakub Voracek. The Flyers went 0-for-3 with the man advantage.
A couple of minutes after the power play ended, the Flyers took advantage of a Nashville turnover as Giroux scored on a slap shot that went up into the net over Mazanec at 16:49.
The Predators answered right back a minute later when Fisher's snap shot from the left circle beat Mason five-hole.
Giroux hit the post during the Flyers' second power play early in the third period, and Nashville killed it off.
Then Wilson put a backhand under Mason's left leg at 5:05 to put Nashville ahead 2-1. Couturier tied it up when Andrej Meszaros found him in front at 13:06.
Notes: The Flyers are 1-1 in the shootout this season and 11-3-1 when scoring first. ... Fisher has three goals in four games, and Nashville is 4-2-0 when he scores this season. ... Wilson's goal was the 50th of his career. ... Cullen played his 1,100th career game.