Columbus owned a 1-0 lead heading into the third, courtesy of a second-period goal from Jeff Carter. Davis Drewiske tied it on a slap shot 4:33 into the closing period and Brown produced the game-winner by tipping in a shot from Drew Doughty.
Drewiske, a defenseman who has been a healthy scratch for most of the season, scored his first goal of the season and just the second of his 102-game career.
Jonathan Quick saved 24 shots to get the win. The Kings pulled out the victory despite failing to produce more than two goals for the 10th straight game.
Curtis Sanford stopped 39 shots for the Blue Jackets.
The Kings, Columbus, Carolina and New York Islanders began the evening tied for last place in the NHL with nine wins each. Los Angeles and Carolina both won while Columbus and the Islanders failed in their bid to reach 10 victories.