Jermaine Johnson gave the home side a first-half lead but goals after the break from from Daniel Nardiello and Troy Archibald-Henville gave City victory.
Johnson struck in the 13th minute, beating a couple of players before firing into the top right-hand corner from the edge of the area.
Neil Mellor then had an effort cleared off the line before Wednesday had a let-off when Jon Otsemobor nearly headed into his own net.
Exeter went close to equalising early in the second-half when Nardiello’s shot was deflected onto a post. Nardiello did find the net in the 76th minute with a low, angled shot from the right after receiving the ball from James Dunne.
Archibald-Henville scored with a fine solo effort a minute from time, collecting the ball from Darren Potter’s stray pass in his own half before striding purposefully into Wednesday’s area and slotting past Nicky Weaver.