After Ottawa goalie Craig Anderson had stopped Matt Read and Claude Giroux in the tiebreaker's first two rounds, Spezza used a series of backhand-to-forehand fakes against Steve Mason before sliding a shot past his stick.
Sean Couturier hit the crossbar on the Flyers' final try.
Clarke MacArthur netted a pair of regulation goals for the Sens, who snapped a two-game losing streak.
Patrick Wiercioch and Kyle Turris also scored and Anderson finished with 25 saves.
Mason turned aside 30 shots.
Celebrity Families of 2014 [PHOTOS]