Lewis Miley, Anthony Gordon and William Osula were on target as the Magpies came from behind to beat the Ravens at St. James' Park on Saturday.
The draw for the next round took place shortly afterwards and United were handed a tie at St. Andrew's @ Knighthead Park against the current League One leaders.
Newcastle and Birmingham have been drawn together in the competition on three previous occasions - with each tie going to a replay.
In 1905/06, United ran out 3-0 winners on Tyneside at the quarter-final stage after a 2-2 draw at St. Andrew's, but the Blues came out on top in 2006/07. The sides drew 2-2 at Birmingham in the third round, but City were 5-1 winners in the replay at St. James' Park.
Then, in 2016/17, again at the third round stage, the teams drew 1-1 at St. Andrew's before United prevailed with a 3-1 win in the replay at Gallowgate.
Ties in the fourth round will be played across the weekend of Saturday, 8th February 2025. More information, including ticket details, will be confirmed on newcastleunited.com in due course.
The full draw is:
Manchester United v Leicester City
Leeds United v Millwall or Dagenham & Redbridge
Brighton & Hove Albion v Chelsea
Preston North End or Charlton Athletic v Wycombe Wanderers
Exeter City v Nottingham Forest
Coventry City v Ipswich Town
Blackburn Rovers v Wolverhampton Wanderers
Mansfield Town or Wigan Athletic v Fulham
Birmingham City v Newcastle United
Plymouth Argyle v Liverpool
Everton v Bournemouth
Aston Villa v Tottenham Hotspur
Southampton v Burnley
Leyton Orient or Derby County v Manchester City
Doncaster Rovers v Crystal Palace
Stoke City v Cardiff City