Derek MacKenzie scored the game-winning goal 0 2, on Monday.
Ryan Johansen, Derick Brassard and Derek Dorsett also scored and Curtis Sanford made 21 saves for the Blue Jackets, who bounced back from a 2-1 loss to the San Jose Sharks on Saturday.
Anton Lander and Ben Eager lit the lamp for the Oilers, who have lost 13 of their last 16 games. Devan Dubnyk gave up three goals on 24 shots in defeat.
Oilers star forward Taylor Hall was injured in pre-game warmups when he was cut on the forehead by the skate of teammate Corey Potter. He was taken to the dressing room and did not play in the game.