St. Louis captured its third straight win, having begun the night in the eighth spot in the Western Conference playoff race.
Backes scored on a backhand shot while sliding on the ice, making his fifth goal of the season a spectacular one.
St. Louis nursed a one-goal lead until Patrik Berglund scored into an empty net with a second remaining in the third period.
Columbus had captured three in a row and a win over the Blues would have moved the Blue Jackets into a playoff position.
Jake Allen made 26 saves to get the victory. Sergei Bobrovsky stopped 16 shots for the Blue Jackets.