David Moyes’ men won 1-0 at Old Trafford on Wednesday night and, moments after the final whistle, were handed another home tie against Barclays Premier League opposition.
Moyes said: "You really only want home draws if you can get them. Norwich are a good side with a good manager, so we will pay them every respect."
The Reds will face Chris Hughton’s Canaries, who beat Watford in extra time in the fourth round on Tuesday, in the week commencing 28 October.
Fourth-round ties are as follows:
Sunderland v Southampton
Leicester City v Fulham
Birmingham City v Stoke City
Manchester United v Norwich City
Burnley v West Ham United
West Brom/Arsenal v Chelsea
Tottenham Hotspur v Hull City
Newcastle United v Manchester City
Ties played week commencing 28 Oct.