Chicago spotted the Blues a 2-0 lead in the first period, but then came roaring back, scoring four straight goals in the middle frame to take control and cruise to a second straight win.
Viktor Stalberg, Dave Bolland, Kane and Toews all found the net the second period and Hossa added an empty-netter for the Blackhawks, whose hopes for a Western Conference playoff spot brightened with their first win St. Louis in four tries.
Corey Crawford turned aside 31-of-34 shots to earn the victory.
Andy McDonald, Brad Boyles and Alex Pietrangelo scored for the Blues.
Ben Bishop was chased in the second period after yielding four goals on 18 shots and Ty Conklin relieved, stopping all seven Chicago shots the rest of the way.