The Bees were an obvious threat from set pieces and John Akinde gave the home side a 12th-minute lead with a thumping header from Ryan Watson's corner.
But Jake Taylor fired in a terrific shot from 25 yards that arrowed into Jamie Stephens' top corner to draw Exeter level in the 43rd minute.
And they were ahead almost immediately after the break, Reuben Reid finding Lee Holmes 10 yards out with a clever back-heeled pass.
The home side looked stunned and the Grecians added a third six minutes later when Stephens spilled an Ollie Watkins shot and David Wheeler followed in.
Barnet almost pulled one back with 20 minutes remaining but substitute Jean-Louis Akpa Akpro saw his shot cleared off the line by Watkins.
And their fate was sealed by Reid's 81st-minute penalty after Mauro Vilhete tripped Taylor in the corner of the box.