Bobrovsky stopped 40-of-43 shots and made scintillating glove saves on Mats Zuccarello and Rick Nash in the tiebreaker, marking an effective return after a 14-game absence with a strained groin.
The Blue Jackets held a 3-1 advantage after two periods but surrendered the lead as New York's Brian Boyle and Nash scored 2 minutes apart early in the final period.
After the Rangers survived a 4-on-3 Columbus power play during a scoreless overtime, the game moved to a shootout.
Mark Letestu bested Henrik Lundqvist in the first round and Ryan Johansen got a backhander past him in the third, hoisting the Blue Jackets to their second win in three games.
Brandon Dubinsky and Jason Wisniewski each notched a goal and an assist, and Cam Atkinson also scored for Columbus.
Nash finished with two goals and Lundqvist made 35 saves for New York.
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