The Premier League side took the lead on 21 minutes when teenager Brandon Mason's low cross was tapped home by midfielder Christian Kabasele for his second goal of the season.
It was comfortable for the Hornets in the first half and they had chances to increase their lead through Odion Ighalo, who blazed over the bar, and Etienne Capoue who saw his shot saved.
The away side offered very little in response and didn't have a single shot on target, the closest they came was when Jamie Ward saw his shot blocked by the home defence.
It was game over 13 minutes from time when Jerome Sinclair, a summer signing from Liverpool, ran at the Burton defence before coolly slotting home for his first goal for the club.
Nigel Clough's side couldn't respond as the Premier League side ended a run of five games without a win as they progressed to the fourth round.