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 Tuesday's heroics, the Blues are still considered rank outsiders to regain the title they won in 2011, with United amongst the favourites after drawing Norwich.
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 Championship outfit Burnley.
Martin Jol's Fulham travel to Leicester in search of a quarter-final berth in their tie, due to be played on October 29 or 30.