Javier Hernandez's goal has set up another home tie for the Reds, to be played on 29/30 October, sandwiched between games against Stoke City and Fulham. United do not play Norwich in the league until to the end of the year.
Of the draw, manager David Moyes said: "You only want a home draw. Norwich are a good side and we'll pay them every respect."
The Canaries needed extra time to see off Watford on Tuesday night, winning 3-2.