The shutout was Philadelphia's league-leading fourth this season. Reserve goaltender Michal Neuvirth has Philadelphia's other three shutouts.
Sean Couturier also scored for the Flyers, who have won two straight. It was the Flyers' first win at Madison Square Garden since Feb. 20, 2011.
Following a strong start to the season, New York has dropped three in a row.
Antti Raanta made 31 saves for the Rangers.
Simmonds' fourth goal of the season opened the scoring 12:24 into the second period.
After Jesper Fast missed on an attempted tip, Philadelphia counterattacked with Matt Read carrying the puck up the ice. Read sent a diagonal cross-ice pass to Simmonds, who skated by Kevin Klein before wristing a quick shot past Raanta.
The goal was Simmonds' first point since Nov. 14.
Philadelphia's lead grew to 2-0 at 3:06 of the third period as Couturier jammed a rebound past Raanta. Couturier's goal was his second of the season. Simmonds added an empty netter in the final minute.
Despite one of its best periods of the season, New York entered the first intermission in scoreless draw. The Rangers outshot the Flyers 10-4 in the first, and played with significant energy.
Dylan McIlrath was a source of that energy for the Rangers.
New York surged when McIlrath hit Nick Schultz in the offensive zone, then had a fight with Luke Schenn. The hit caused Schultz to miss the remainder of the game with an upper body injury.
For coming to the aid of his teammate, Schenn was assessed a minor for instigating, a major for fighting and a 10-minute misconduct.
The Rangers fired seven shots on goal following McIlrath's hit and subsequent fight, but New York couldn't break through against Mason. The goaltender's biggest save in the period was a lunging stop against Dominic Moore.
New York was unable to generate much after the first period, as Mason only had to make 14 saves in the final 40 minutes. The Flyers finished the game outshooting New York 34-24.
NOTES: New York played its first game without center Derek Stepan, who suffered two broken ribs in the Rangers' 4-3 loss to the Boston Bruins on Friday. New York coach Alain Vigneault reinserted Emerson Etem and Viktor Stalberg into the lineup, and had Kevin Hayes center a second line of Etem and Chris Kreider. ... Vigneault said New York would "have a better idea" Sunday how long Stepan will be out. He was scheduled to see a doctor Saturday. ... Philadelphia scratched Sam Gagner, Vincent Lecavalier and Brandon Manning. Dan Boyle was New York's lone scratch.