P.A. Parenteau was whistled for hooking 42 seconds into the extra frame, and on the ensuing 4-on-3 power play, Anisimov got control of the puck in the right circle and used a screen from R.J. Umberger to fire the game-winner past Semyon Varlamov.
Vinny Prospal's man-advantage tally early in the third period had tied the contest.
Sergei Bobrovsky turned aside 18-of-19 shots for Columbus, which broke a four-game losing streak.
Parenteau scored and Varlamov finished with 32 saves for the Avalanche, who fell for the fourth time in five contests.