The victory made it four games unbeaten on the road as promotion-chasing hosts suffered a slump in form.
The Shots took Burton by surprise by grabbing the lead in the fourth minute when on-loan Dani Lopez spurted onto Danny Hylton's flick to bury a low shot into the bottom corner of the net from 20 yards.
Visiting keeper Jamie Young had to get his fingers to a raking low drive from Chris Palmer as the hosts tried to rally, but the Shots looked dangerous on the break and Peter Vincenti was not far off target with a cleverly curled shot.
Young had to paw a Matt Paterson effort to safety but the Shots were good value for their half-time lead.
Hylton went close to increasing Aldershot's lead with fierce low shot in the 57th minute and Lopez forced a point-blank save from Burton keeper Dean Lyness six minutes later. Young had to make a rare save to keep out a Billy Kee shot as Burton searched in vain for the equaliser.