Eric Brewer also lit the lamp and Chris Mason stopped 32-of-34 shots for the Blues, who won their third straight and kept their faint Western Conference playoff hopes alive by moving to within six points of idle eighth-place Colorado with six games remaining.
Moments after Mason stopped the Blackhawks' Marian Hossa on a breakaway early in the second period, Brewer responded by finding the net to forge a 2-2 tie.
Steen gave the Blues a crucial one-goal edge at 3:27 of the third when he poked in a David Backes pass.
Patrick Sharp and Hossa connected for Chicago, losers of four of five. The Hawks, however, were able to maintain their five-point lead over Nashville in the Central Division after the Preds were defeated by Los Angeles.