Dennis Rasmussen, Brian Campbell, Niklas Hjalmarsson and Artemi Panarin also scored for the Blackhawks, who have won have four straight. Scott Darling made 23 saves to improve to 9-2-2.
Patrik Berglund scored twice for the Blues, who had recorded at least one point in each of their last 14 home games. Alex Pietrangelo and Kyle Brodziak also scored for the Blues, who fell to 13-2-3 at home.
Hjalmarsson started the comeback by scoring in the first minute of third period to tie the score at 4-4.
Hinostroza converted with 4:25 left for his first game-winning goal of the season. Panarin added an empty-netter.
Kane tied it 2-2 midway through the second period with his first goal since Nov. 26.
Chicago centre Artem Anisimov left in the first period and did not return.
The two teams will next meet in the Winter Classic on Jan. 2 at Busch Stadium
NOTES: Chicago G Corey Crawford missed his eighth straight game after undergoing an appendectomy on Dec. 3. He skated for the first time on Thursday. … Chicago, St. Louis and Buffalo are the only teams in the NHL yet to score a short-handed goal this season. … St. Louis D Brad Hunt, a healthy scratch, has four points in four games since being recalled on Dec. 8.
Blackhawks: Host San Jose on Sunday in the first of four straight home games.
Blues: Host Edmonton on Monday in the finale of a three-game homestand.