Sissons got his second goal of the season early in the third, giving the Western Conference champions their fourth win in five games. Rinne was strong in his 44th career shutout and first of the season.
The Flyers had combined for 13 goals in their first two home games and scored 26 overall in their first six games.
Philadelphia pulled goalie Michal Neuvirth on a power play and couldn't score in the final minute with a 6-on-4 advantage.
The Flyers had outscored the opposition 12-4 in the second period, and their revitalized offense a year after missing the playoffs gives hope that better days are ahead.
But they couldn't solve Rinne.
The Predators beat the Flyers 6-5 nine days ago, but the rematch was far more defensive. The Flyers and Predators were locked in a scoreless tie through two periods, each with 16 shots.
Turned out, the Predators just needed Sissons in the lineup — he sat out the game last week with an injury.
Wayne Simmonds, who leads the Flyers with five goals, briefly left the game late in the second when he appeared to get poked in the face with a stick. Simmonds had proclaimed himself healthy at morning skate after he left a win against Florida on Tuesday with an undisclosed injury.
Neuvirth earned a second straight start on the heels of his 40-save effort against the Florida Panthers on Tuesday night. Brian Elliott has three wins for the Flyers, though neither goalie has established himself as a true No. 1.
The Flyers lost on the 50th anniversary of their first home game, a 1-0 win over the Pittsburgh Penguins at the Spectrum. The win was the first of 696 Flyers' victories at the since-demolished building and home of the Broad Street Bullies.
The Flyers honored founder Ed Snider before the game with a 9-foot bronze statue outside the Wells Fargo Center. Snider remained chairman until his death in April 2016.
Snider's statue has a Stanley Cup championship ring on his left ring finger that fans are encouraged to rub for good luck. Flyers President Paul Holmgren was one of the first to rub the ring on the statue.
NHL Commissioner Gary Bettman praised Snider's love and dedication of the Flyers and Philadelphia. Dozens of former team greats attended the ceremony.
"Everything I am as a human being, thank you Ed Snider," Hall of fame goalie Bernie Parent said as he threw a kiss toward the statue.
