The Magpies' second-string booked their last-eight place with a 3-2 win over AFC Bournemouth and will play away to either the Blades or the Rams in the next round.
The two teams kick off at Stocksbridge Park Steels' Bracken Moor ground at 7pm (BST).
Quarter-final dates will be confirmed in due course, but they are scheduled to be played between Friday, 6th April and Monday, 9th April.