Columbus ended a three-game skid in a contentious contest that included two fights and 19 penalties.
Derek MacKenzie opened the scoring for the Blue Jackets with 7:42 to go in the first period and Johansen then put a slap shot past Braden Holtby.
Second-period goals from Brandon Dubinsky and Johansen boosted the Columbus lead. Washington fought back within two, but Cam Atkinson closed the scoring for the Blue Jackets 4:22 into the third.
The teams began the night tied for fifth place in the Metropolitan Division and were one point outside a playoff position.
Sergei Bobrovsky stopped 30 shots to claim the win. Holtby allowed five goals on 29 shots.