Neither team could score during the first four shootout rounds and then both sides managed a goal in the fifth and sixth rounds.
The Flyers forced the shootout and wound up gaining a point thanks to a power-play goal by Simmonds with 1:24 to go in the third period.
Shea Walker had put Nashville in front at 3-2 early in the third.
Philadelphia lost for the third time in four games after having won seven out of eight.
Hutton finished with 27 saves before stopping 5-of-7 shootout attempts. Mason was credited with 34 saves.