Philadelphia was facing the possibility of losing four games in regulation to open the season, something the Flyers have never done.
Simmonds broke a scoreless tie by tipping in a shot with 8:07 to go in the second and Voracek scored on a wrist shot with 4:06 remaining.
Bryzgalov surrendered a goal to Taylor Pyatt just 1:39 into the third, but the Flyers held on to hand the Rangers their third loss in four games.
New York had defeated Philadelphia in each of their previous eight meetings.
New York goaltender Henrik Lundqvist had 31 saves.