Mason did not allow a goal in three shootout attempts, and he forced the shootout by stopping Colin Wilson on a breakaway with 34 seconds left in the overtime.
The Flyers were outshot, 36-24, but Mason was the difference.
"He did a great job once again," Couturier said. "Since the start of the year he's been unbelievable. He's making key saves at the right time and keeping us in games when we're not playing at our best."
Mason made his best shootout save when he stuck out his left leg at the last minute to stop Matt Hendricks.
"I bit hard on his first move and was kind of down and out," Mason said, "but those are the battling type of saves I try to work on in practice."
Vinny Lecavalier, on the Flyers' first attempt, scored what proved to be the winner in the shootout. Matt Cullen fired wide on the Predators' last attempt.
Couturier converted a slick pass from Andrej Meszaros to tie the game at 2 with 6 minutes, 53 seconds left in regulation.
Jake Voracek had an open net with a little over a minute left in regulation but couldn't handle a bouncing puck out front. David Legwand had a golden chance for the Predators, but Mason stopped him with 26 seconds to go.
The Predators had taken a 2-1 lead as Wilson raced past Voracek and put a backhander through the legs of Mason with 14:55 remaining in the third period.
Nashville was playing without two injured game-changers: defenseman Shea Weber (eye) and goalie Pekka Rinne (hip). Earlier this year, the Flyers lost to Washington and Tampa Bay when they were playing without injured superstars Alex Ovechkin and Steven Stamkos, respectively.
Injuries to Weber and veteran Kevin Klein meant that Nashville was playing with a defense that had played a combined 305 NHL games before Saturday.
"Usually when you have your best guy out, the team is playing even harder," Meszaros warned before the game. "And the other team is not really focused, so we have to be ready for that."
Yet, the Preds controlled the first period. The Flyers were sloppy and second to the puck for most of the first 20 minutes, which ended tied at 1. It was tied only because Mason made a handful of outstanding saves after Flyers defensive breakdowns.
Exactly a minute after Giroux's first-period goal, sloppy play by Braydon Coburn and Steve Downie enabled Mike Fisher to tie it with a wrist shot from the left circle.
Couturier, who was brilliant on the penalty-killing unit, and Matt Read had a two-on-one shorthanded chance late in the second period, but rookie defenseman Seth Jones broke it up.
Nashville finished 0 for 5 on the power play.
"This is huge to start the road trip on the right step," Giroux said.