Clarkson had the Devils' goal. Johan Hedberg made 21 saves.
It's the first time the Flyers have won this season when tied after two periods (1-5) and their first shootout win in more than a year (March 10, 2012).
The Devils defeated the Flyers 5-2 Wednesday in Newark.
Each team had multiple chances to win Friday in regulation, with the Devils coming closest after Flyers forward Brayden Schenn was called for charging Peter Harrold with 1:42 remaining in the third period.
A shot from the point hit New Jersey's Steve Bernier in front and trickled just wide of the goal with 12 seconds remaining.
The Devils got the best of play in overtime, but Bryzgalov made three outstanding saves with 1:52 left, some of them from his back. He stopped Elias twice in the slot, the second time while down. Marek Zidlicky spotted the puck lying loose between Bryzgalov's skates and tried lifting a shot over the netminder, but Bryzgalov got his toe on it, it fluttered down next to his arm, and he was able to trap it against his body.
The Flyers' penalty killers came up big all game, denying all five New Jersey chances, including a pair in the third period. The biggest kill came midway through the first period when the Flyers' advantage turned into a disadvantage when Wayne Simmonds and Jakub Voracek were sent off for minor penalties at 10:34. The Devils had a 28-second 4-on-3 and a 1:32 5-on-3, but the Flyers were able to survive both.
Philadelphia also shut down New Jersey on a power play at 8:51 of the third when Scott Hartnell was sent off for slashing.
New Jersey's penalty killing also stepped up, including a two-minute Flyers advantage 15 seconds into the third period when Adam Henrique was sent off for high-sticking Flyers captain Claude Giroux. The Devils also held the Flyers down when Anton Volchenkov was sent off for interference at 13:12.
Clarkson tied the game at 6:09 of the second period. Bryce Salvador fired a shot from the left point that Clarkson, posting up Flyers defenseman Luke Schenn in the slot, tipped past Bryzgalov for his 11th goal of the season, his second in 16 games.
Talbot gave the Flyers the lead at 15:21 of the first. He led a rush up the ice and skated to the net. Matt Read passed across to Simon Gagne, who skated through the slot and found an open passing lane when Devils defenseman Andy Greene collided with Talbot and fell into Hedberg, taking them both out of the play. Gagne got the puck to Talbot on the right post, and he scored his third goal of the season into an empty net.
"We had a 3-on-2 and [Read] made a great play to [Gagne]," Talbot said. "[Gagne] took the goalie with him and gave the puck to me back door and I shot it and put it in."
Contact Adam Kimelman at email@example.com. Follow him on Twitter: @NHLAdamK