City moved through to the last four with a shoot-out win over Leicester on Tuesday.
League One side Burton are currently 15th in the table and booked their place in the semi-final with a 1-0 win away at Middlesbrough this week.
The first leg is set to be played the week commencing Monday 7 January and the second leg the week commencing Monday 21 January, with exact date and times to be confirmed in the coming days.
The Blues are looking to win the competition for the fourth time in six seasons.
City and Burton Albion have never met competitively in any competition.
Brewers manager Nigel Clough played for the Blues from 1996 to 1998, playing 39 games and scoring four goals.
Former EDS striker Devante Cole plays for Burton.
Spurs will take on Chelsea in the other semi-final.