The Jets were holding a 2-1 lead midway through the final period when Kane broke into the Blue Jackets zone as part of a 3-on-1 rush. He kept the puck and fired a wrist shot that found the far corner of the net for the decisive goal.
Kane also had an assist in his return from a six-game absence with a lower-body injury.
Bryan Little and Mark Scheifele also scored as Winnipeg broke a three-game losing streak.
Al Montoya made 22 saves.
Fedor Tyutin and R.J. Umberger each had a goal for Columbus.
Mike McKenna turned aside 29 shots.