The Blues beat the Canaries following a penalty shoot-out in Wednesday night's third round replay at Stamford Bridge.
After kick-off was delayed due to travel disruption around the ground, Michy Batshuayi gave Chelsea a 55th-minute lead but Jamal Lewis levelled in the fourth minute of stoppage time.
In extra time, Pedro and Alvaro Morata were both sent off for the home side but there were no further goals.
However, Chelsea won 5-3 on penalties and that means United - who beat Luton Town 3-1 in the third round - will face Antonio Conte's side on Sunday, 28th January for a televised 1.30pm (UK time) kick-off.
Ticket details will be confirmed on nufc.co.uk on Thursday morning.