Marcus Stewart gave Exeter a first-half lead before Aldershot took control after the break with two goals in five minutes from Jake Robinson and Ben Harding.
But with ten minutes left, Harding headed the ball into his own net and Neil Saunders curled an unstoppable free-kick into the top corner for the Grecians.
Aldershot started brightly and had the first chance of the game in the second minute when Marvin Morgan forced Paul Jones to save well with a shot from close range.
Harding struck over the bar and Louis Soares flashed a shot across the face of goal for the Shots as they increasingly looked like opening the scoring.
But Exeter took the lead on the half-hour mark when Stewart stroked the ball home after he was found by Adam Stansfield.
Exeter ended the half strongly and Saunders was unlucky not to get on to the end of a sweeping move involving Stansfield and Richard Logan.
After the break, Danny Seaborne headed over from a Saunders corner and Logan forced Alex McCarthy to make a fine save as Exeter tried to put the game beyond Aldershot.
But the Shots were level ten minutes after the restart. Harding played a long ball forward to Morgan who found Robinson with a neat ball and the striker buried it past Jones.
Harding then beat Jones with a long-range effort which bounced over the diving goalkeeper.
Exeter pushed forward and their reward came with two goals in two minutes.
First Harding glanced the ball off the top of his head and into his own net from Matt Gill's long throw.
The crowd were still celebrating when Saunders curled a fantastic free-kick into the top corner from 20 yards to win the game.