Edmonton faced a 2-1 deficit heading into the final 20 minutes, but Jones erupted for his seventh, eighth and ninth goals of the season -- the last an empty-netter with 38 seconds remaining.
Ladislav Smid and Ales Hemsky also scored for the Oilers in the final period, giving them their first win in four games.
The Blue Jackets had been trying to put together back-to-back wins for just the second time this year. They have only seven victories on the season, having opened the campaign with eight consecutive losses.
Devan Dubnyk gave up a goal in each period, but made 36 saves to get the win. Curtis Sanford allowed five goals on 33 shots.