Javier Hernandez's strike early in the second half secured a 1-0 win at Old Trafford, which will play host to the Canaries the week starting October 28.
Birmingham ousted holders Swansea 3-1 and have been handed another home match in the last 16, this time against Mark Hughes' Stoke.
Despite their heroics, the Blues are still considered rank outsiders to regain the title they won in 2011, with United amongst the favourites after drawing Norwich.
"Well, you always want a home draw if you can get it,'' manager David Moyes told Sky Sports 1.
"Norwich are a good side, they have got a good manager and we will give them every respect.''
Elsewhere, Newcastle will welcome Manchester City to St James' Park after overcoming Leeds, while Hull travel to Tottenham just days after heading to White Hart Lane in the Premier League.
Managerless Sunderland play host to Mauricio Pochettino's Southampton, while West Ham head north to take on Sky Bet Championship outfit Burnley.
Martin Jol's Fulham travel to Leicester in search of a quarter-final berth in their tie.
The last tie to be decided is arguably the most exciting of the fourth round, with Arsenal hosting Chelsea after Arsene Wenger's side overcame West Brom on penalties on Wednesday night.
Draw in full:
Sunderland v Southampton
Leicester v Fulham
Birmingham v Stoke
Man Utd v Norwich
Burnley v West Ham
Arsenal v Chelsea
Tottenham v Hull
Newcastle v Man City
Ties to be played on October 29 and 30.