The draw was made on Sky Sports News earlier this morning by former Arsenal forward Paul Merson and split into Northern and Southern sections.
It will be Aitor Karanka's first competitive game in charge of Blues since taking over as Head Coach earlier this month.
United, who are coached by Mark Bonner, finished last season's league campaign in 16th place.
Blues lost out to Portsmouth in round one of the League Cup last season, when a young side were beaten 3-0 at Fratton Park.
Meanwhile, the U's suffered a 6-0 away defeat at Swansea City in round two.
Blues have twice lifted the League Cup, back in 1963 and 2011.
The current holders of the trophy are Manchester City who beat Aston Villa in the final at Wembley in March 2020.
The first round tie between Blues and Cambridge is scheduled to be played at the St. Andrew's Trillion Trophy Stadium over the weekend of 5-6 September 2020. However, clubs are able to mutually agree an alternative date prior to the start of the international window which commences on 31 August.