Chernyshov gave San Jose a 3-2 lead when he buried a cross-ice pass from Will Smith with a one-timer from the right face-off circle to the blocker side on Elvis Merzlikins.
Macklin Celebrini had a goal and an assist, and Smith had two assists for the Sharks (33-31-7), who had been 0-5-1 during the streak. Alex Nedeljkovic made 22 saves.
Denton Mateychuk and Cole Sillinger scored, and Merzlikins made 33 saves for the Blue Jackets (38-24-11), who have lost two straight and three of four.
Mateychuk gave the Blue Jackets a 1-0 lead just 31 seconds into the first period with a wrist shot from the left circle past a screened Nedeljkovic to the blocker side.
Chernyshov tied it 1-1 at 17:17, taking a pass from Smith and putting a snap shot from the right circle off Merzlikins' blocker and into the net.
Sillinger put Columbus up 2-1 at 12:09. Ivan Provorov intercepted a clearing attempt from Sharks defenseman Dmitry Orlov and fed a cross-slot pass to Sillinger in the right circle, where he dropped to one knee and beat Nedeljkovic with a one-timer.
Celebrini tied the game 2-2 just 57 seconds into the third period with a power-play goal. Following a face-off win by Alex Wennberg in the left circle, Orlov gained control of the puck and passed to Celebrini, who beat Merzlikins to the glove side with a one-timer off the right post.
Blue Jackets forward Dmitri Voronkov left the game at 2:18 of the second period with an upper-body injury after blocking a shot by Mario Ferraro off his right hand.
