Philadelphia won for the seventh time in 10 games and inched closer to wrapping up a post-season berth.
Simmonds beat Braden Holtby with a wrist shot for what turned out to be the winning goal. Ilya Bryzgalov then stopped Washington's Troy Brouwer.
Matt Read scored in the shootout's first round for the Flyers and Matt Hendricks promptly responded for the Capitals, who picked up a critical point in their bid to make the playoffs.
Washington is in the eighth and final playoff position in the Eastern Conference, one point in front of Buffalo.
Bryzgalov made 30 saves in the win. Holtby was credited with stopping 27-of-28 shots.