Dave Watson's young Magpies will travel to Carrow Road for a 7pm (UK time) kick-off on Wednesday, 7th February.
The Magpies beat Brighton & Hove Albion 5-0 in the third round before edging past Crystal Palace in the fourth round with a 3-2 extra-time victory.
Norwich, meanwhile, were 4-1 winners against Barnsley in the third round and saw off Derby in the fourth round, with the final score 4-2.
A Canaries side including Jacob Murphy won the competition in 2013. Newcastle last lifted the trophy in 1985.