Fulham had to come back from behind to beat City after Jordi Hiwula scored in the 20th minute to give the home side the lead.
Cauley Woodrow equalised for Fulham on 65 minutes and the same player struck again with eight minutes remaining to give them the tie.
Full results from 5th round
Southampton 1 Nottingham Forest 5
Swansea City 1 Manchester United 5
Blackburn Rovers 4 Stoke City 0
West Bromwich Albion 1 Burnley 2
Newcastle United 2 Queens Park Rangers 1
Manchester City 1 Fulham 2
Chelsea 3 West Ham United 3 - Chelsea won 5-4 on pens
Charlton Athletic v Tottenham Hotspur - to be played tomorrow
Sixth Round Draw
Fulham v Burnley
Nottingham Forest v Chelsea
Newcastle United v Blackburn Rovers
Manchester United v Charlton Athletic or Tottenham Hotspur