The 40-year-old leaves the League Two Brewers after five-and-a-half years as assistant and manager to join the Blues in the Championship.
"Gary is an ambitious young manager and the offer from one of his former clubs playing Championship football was one he wanted to take," Burton chairman Ben Robinson told the Brewers' website.
Rowett's task this season will be to keep the Blues in the second tier. After 14 games of the season they are second bottom and on a run of three straight defeats.
On Saturday they were thrashed 8-0 at home to Bournemouth and next face high-flying local rivals Wolverhampton Wanderers at the weekend.