Phil Parkinson's side caused one of the greatest FA Cup upsets when it came from two goals down to win 4-2 at the Premier League leader.
And top-flight side Sunderland and second-tier outfit Fulham, who replay next week after a goalless draw, will both be wary of suffering the same fate as Jose Mourinho's men after the fifth-round draw.
A League Cup finalist two years ago, Bradford City last faced Sunderland in a 3-0 defeat in December 2003, while it has to go back to 1994 to find its last meeting with Fulham, which ended in a goalless draw.
If Manchester United can get past fourth-tier Cambridge in its replay, it will travel to the winner of Preston North End and Sheffield United, while holder Arsenal hosts Middlesbrough, the Championship high-flyer which sprang a shock of its own with a 2-0 win at Manchester City.
Liverpool and Bolton Wanderers will replay for the right to go to Crystal Palace, while Leicester City and Aston Villa meet in an all-Premier League affair.
West Bromwich Albion faces West Ham United at home, while Stoke City - a 4-1 winner at Rochdale - will travel to Blackburn Rovers, which enjoyed a fine victory over Swansea City in the last round.
Derby plays host to Reading in an all-Championship tie.
FA Cup fifth-round draw:
Crystal Palace v Liverpool or Bolton Wanderers
Arsenal v Middlesbrough
Aston Villa v Leicester City
West Bromwich Albion v West Ham United
Bradford City v Sunderland or Fulham
Blackburn Rovers v Stoke City
Derby County v Reading
Preston North End or Sheffield United v Cambridge United or Manchester United
The ties will take place over the weekend of 14-15 February.