Jonathan Toews scored in the first round of the tiebreaker, and Brad Marchand answered in the top of the third round to keep the contest alive.
Kane followed by coming in slowly on Bruins goalie Tuukka Rask and using a series of fakes before firing a shot to his stick side for the game-winner.
Marian Hossa and Brandon Bollig scored in regulation for the Central Division-leading Blackhawks, who took a third win four games.
Corey Crawford stopped 34-of-36 shots to earn the win.
Marchand scored twice in regulation for Atlantic Division-leading Boston, which fell for the fourth time in six contests.
Rask finished with 35 saves.