The Blackhawks spotted St. Louis a 2-1 lead, but Hossa's score-tying, shorthanded tally with 7:49 remaining the second period ignited a scoring binge that allowed Chicago to win going away.
Patrick Sharp scored on a power play with 5 minutes left in the middle frame, handing Chicago its first lead, and Hossa made it 4-2 early in the third on another power play.
Ray Emery stopped 23-of-25 shots to earn the victory.
David Perron returned from a 13-month absence with a concussion and scored in the first period for the Blues, who fell to 8-2-3 in the their last 13 games.
Chris Stewart also scored and Brian Elliot made 31 saves for St. Louis.