Backstrom beat Josh Harding with a wrist shot to the glove side after the first four skaters failed to score in the shootout.
Washington pulled out the victory despite trailing deep into the third period. Marcus Johnson scored on a deflection with 3:08 to go in regulation to create overtime and the Capitals went on to win their fourth in a row.
Washington's first regulation goal came from Alex Ovechkin, his 13th of the year.
Braden Holtby made 33 saves to claim the victory. Harding stopped 25 shots.