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.
Millions of Getty images now available for free via embed tool
Ray Liotta sues skin care company over use of likeness