After the first five shooters failed to score in the tiebreaker, Johansen used a series of fakes to get Buffalo's Ryan Miller out of position and then lifted a wrist shot into the net for the game-winner.
Brandon Dubinsky supplied a goal and an assist for the Blue Jackets, who matched the longest winning streak in franchise history.
Matt Calvert and David Savard also scored.
Sergei Bobrovsky turned aside 26 shots and added two more saves in the shootout.
Tyler Myers notched two goals for the Sabres, losers of three straight.
Drew Stafford also scored and Miller finished with 32 saves.