The Predators carried a two-goal lead into the final period, but goals by Gagne and Pronger 3:04 apart tied the game late in the third.
"We've got to play much better with the puck," Nashville coach Barry Trotz said. "There are a lot of areas in our game that we can correct, but getting the two points in the shootout was huge."
Carter opened the scoring at 1:21 of the first period with a wrist shot from the low slot that beat Rinne high to the glove side. It was Carter's 33rd goal of the season. He had 11 shots on goal in the game.
The Predators are now 15-21-4 this season when their opponents score first.
Sullivan tied it just over 3 minutes later when he tapped in a puck just underneath the pad of Philadelphia goalie Michael Leighton.
Erat gave the Predators a 2-1 lead at 7:04 of the first. Dan Hamhuis kept the puck in at the blue line and passed to Jason Arnott on top of the left circle. Arnott sent a backhand pass to Erat, and he beat Leighton with a wrist shot over the glove for his 20th goal of the season.
Arnott has assists in his past three games and points in five of his last six.
"When they tied it up, we did not sit back on our heels," Arnott said. "You can say it was an ugly win, but we will take the two points and that is all that matters."
Leighton left the game just under 3 minutes after the Erat goal with a high ankle sprain. He was replaced by Brian Boucher at 9:45.
"I think every goalie is prepared just in case there is a slow start by the team or anything like that," Boucher said. "It is a little easier when it's that early; you are not sitting as long."
The Predators put immediate pressure on the new Flyers goalie, scoring a goal just 20 seconds after he entered the game.
Boucher stopped Joel Ward's shot from the right faceoff dot, but Smithson was parked just outside the crease and backhanded the rebound in for his ninth goal of the season.
"We stayed composed and positive," Smithson said. "Those are signs of a good team."
"The first period the score is 3-1," Philadelphia coach Peter Laviolette said. "We outchanced them 2-to-1, outshot them, and we were playing on our toes, but pucks end up in our net."
Gagne got the Flyers within a goal at 11:14 of the third. Pronger's goal came on a wrist shot from the low slot with the Flyers on a power play.
"We had a number of good chances," Pronger said. "In the third period alone, we probably could have had four or five goals. We got a couple to tie it up. Unfortunately for us, we lost in the shootout."
The Flyers last played in Nashville on Dec. 3, 2005. ... Leighton started his 15th consecutive game Tuesday. ... Flyers RW Danny Briere had two assists in the game. He has three points in his past two games The Predators are 4-1-1 vs. Atlantic Division opponents.