The Ducks, 0-3-1 since the Olympic break, got goals from Lubomir Visnovsky and Ryan Getzlaf.
Ducks starting goalie Jonas Hiller gave up three goals on 20 shots before being replaced in the second period by Curtis McElhinney, acquired last week from Calgary for Vesa Toskala. McElhinney blocked seven shots.
Columbus held a 3-0 lead before Visnovsky hit a slapshot from the slot at 9:11 of the third and Getzlaf slapped home a wrister from the left circle at 14:56 to cut the Ducks' deficit to one goal.
But after Saku Koivu was called for hooking with 2:31 left, Vermette scored on a backhander from the top of the crease with 1:58 remaining, and Brassard scored on the power play with 17 seconds remaining.
Columbus went 3-for-6 with the extra skater, while Anaheim failed to score on seven power-play chances.
The Blue Jackets played without all-star forward Rick Nash, who is out with a lower-body injury.
2014: The Year in Music [PHOTOS]