The South Yorkshire duo are among six White Rose sides on the road following a draw made by Ruud Gullit and Paul Ince.
Premier League leaders Manchester City will also face second-tier opposition in Rotherham as do Manchester United, who are at home to Reading.
Shrewsbury, who have appointed Sam Ricketts as boss, have seen off Salford City and Scunthorpe United to reach the third round stage for the fifth time in 23 attempts.
Meanwhile, Cardiff City will travel to League One side Gillingham and Aston Villa will host Swansea City.
Saturday's second round win at Charlton Athletic means Rovers are in the hat as the 44 clubs from the Premier League and Championship enter the competition.
Liverpool coach Juergen Klopp, right, celebrates after Liverpool forward Divock Origi, third from left, scored his side's first goal at the end of the English Premier League soccer match between Liverpool and Everton at Anfield Stadium in Liverpool, England, Sunday, Dec. 2, 2018.
Ties will be played on the weekend of January 5, meaning Rovers' League One clash at home to Southend United will have to be re-scheduled.