The Blues were down 3-1 after the first period but battled back with a power-play goal from Vladimir Tarasenko in the second and an equalizer from Chris Stewart in the third.
Shortly after the start of the extra frame, Backes intercepted a pass from Blue Jackets defender Jack Johnson at the blue line and skated in on goalie Mike McKenna.
With Fydor Tyutin draped over him, he made a backhand-to-forehand move and put a shot past McKenna for the game-winner.
Tarasenko finished with two goals and Jaroslav Halak made 30 saves as St. Louis notched a third straight win.
Tyutin, Artem Anisimov and Ryan Johansen each scored for Columbus.
McKenna finished with 27 saves.