St. Louis has opened the season with a 6-1 record, its best mark to begin a campaign in 15 years.
Jackman scored just 2:19 into the contest and both Patrik Berglund and Vladimir Tarasenko delivered goals in the opening period.
Elliott made 24 saves.
Columbus, a winner just twice in eight outings, began a six-game homestand. Sergei Bobrovsky opened in goal for the Blue Jackets, but was removed after giving up three goals on 11 shots. Steve Mason turned aside all 13 shots he saw in relief.