Mason took the blame for the loss.
"The guys gave me two opportunities to win the game and I didn't do it," Mason said, referring to shootout scores by Brayden Schenn and Sean Couturier. "I let them down tonight."
Mason made 33 saves and kept the Flyers within striking distance with some key third-period stops.
The Flyers have erased third-period deficits eight times to win this season.
Two years ago, mammoth defenseman Shea Weber wanted to be a Flyer. The Flyers wanted him. It seemed like the perfect marriage. But Nashville matched the 14-year, $110 million offer sheet that Weber signed with the Flyers, keeping him with the Predators.
Weber gave a glimpse Thursday of what could have been if he had been playing his home games at the Wells Fargo Center, collecting two assists and a power-play goal.
After winning 10 straight at the Wells Fargo Center, the Flyers have dropped two in a row at home.
Weber scored his ninth power-play goal - it gave the Predators a 3-2 lead with 14:50 to go - while the Flyers were killing a four-minute high-sticking penalty to Nick Grossmann.
With the score tied at 3, Matt Read nearly won it when his shot hit the post and deflected off goalie Carter Hutton's back but somehow stayed out of the net with 1:34 remaining in overtime.
Wayne Simmonds scored a power-play goal with 1:24 left in regulation to knot the score at 3-all. The Flyers had pulled Mason and were on a six-on-four when Scott Hartnell and then Simmonds deflected Kimmo Timonen's point drive into the net.
It gave Simmonds three points for the night and 20 points in his last 15 games.
Andrej Meszaros took a slick pass from Simmonds and fired a left-circle shot past Hutton on the short side, tying the score at 2-2 with 3.8 seconds left in the second.
"I heard him yelling for the puck," Simmonds said. "No one picked him up."
Meszaros also assisted on the Flyers' first goal.
Carter has been in Nashville's goalie rotation because Pekke Rinne, the Predators' standout netminder, has been sidelined because of an infected hip.
A former Phantom who dressed as a Flyers backup goalie for one game, Carter made one of the season's best saves when he robbed Michael Raffl with 12:52 left in the second period.
Carter went behind the net to retrieve the puck, but it caromed crazily off the back boards to Raffl in the slot. But as Raffl shot at the open net, Carter dove back into the picture and made a spectacular glove save.
Raffl was tripped on the play by David Legwand, and the Predators killed the subsequent penalty.
Nashville dominated the second period, outshooting the Flyers, 14-8.
First-period goals by Eric Nystrom (son of Islanders legend Bobby) and Legwand (power play) wiped out an early Flyers lead and put the Predators ahead, 2-1.
A turnover by Raffl enabled Nashville to keep the puck in the zone, and Gabriel Bourque's pass caromed off Nystrom's skate and past Mason to tie the game at 1-1 with 9:03 left in the first. About 51/2 minutes later, Legwand took a feed from Mike Fisher and scored from a one-timer from deep inside the right circle.
Legwand scored on a five-on-four power play, just five seconds after a five-on-three had ended.
Earlier, Schenn tapped in a loose puck after Simmonds' shot trickled free in front. Meszaros made a nice move around a defender to set up the play.