The Blackhawks trailed 2-0 as the game was nearing the midway point of the third stanza, but Kane got the rally started by scoring his 14th goal of the season.
Patrick Sharp tied the contest less than a minute later and Kane produced the game-winner with just18 seconds to go in regulation.
A shot from the point by Niklas Hjalmarsson deflected off a Calgary defender and right to Kane, who put a backhander past Retto Berra.
Antti Raanta made 20 saves to get the win in his first NHL start.
Calgary built its lead on goals by Matt Stajan and Sean Monahan. Berra stopped 28-of-31 shots in the loss.
Chicago won its fourth in a row while he Flames have dropped two out of three.