Steen was the third Blues shooter to score in the tiebreaker, jamming a shot underneath Columbus goalie Steve Mason in the top of the fourth round. Bishop then turned aside a last-chance shot by Antoine Vermette with a left-pad save, handing the Blues a win after four straight losses.
Andy McDonald netted a pair of goals, Patrik Berglund added a goal and an assist and Bishop stopped 19-of-20 shots after replacing an ineffective Ty Conklin in the first period.
T.J Oshie also scored and David Backes added two helpers for St. Louis.
Jakub Voracek had a goal and an assist as the Blue Jackets dropped to 0-4-1 in their last five games.
Matt Calvert, Derek MacKenzie and Fedor Tyutin all found the net and Mason made 32 saves for Columbus.