Winger Alex MacDonald put the home side on their way with the first goal of his loan spell from Burnley before midfielder Robbie Weir's headed double sealed the visitors' fate.
Cheltenham looked the livelier side in the opening 20 minutes only to be rocked by a double salvo as the hosts clicked into gear.
The visitors were guilty of loose defending that gave MacDonald the chance to fire the home side in front from eight yards in the 22nd minute.
Then MacDonald turned provider four minutes later with a cross that gave the unmarked Weir the chance to find the far corner with a 12-yard header.
MacDonald almost laid on Burton's third for Billy Kee before the break, but the striker's shot was blocked by the legs of visiting keeper Scott Brown.
Cheltenham's hopes of clawing their way back into the game were in ruins when Weir met a Kee cross to head his second of the game in the 48th minute.
Byron Harrison pulled a goal back when he headed home a Billy Jones corner in added time.