The Blades and the Rams meet at Stocksbridge Park Steels' Bracken Moor ground on Tuesday, 27th March.
Newcastle booked their place in the last eight with a dramatic 3-2 win over AFC Bournemouth on Friday afternoon.
Quarter-final dates will be confirmed in due course, but they are scheduled to be played between Friday, 6th April and Monday, 9th April.
As with the Round of 16, ties will be one-off matches, decided by extra-time and, if required, penalties if they are level after 90 minutes.
While United have struggled in Premier League 2 this season, they've fared much better in cup competitions. As well as reaching the last eight of the Premier League Cup, they will face Morpeth Town the final of the Techflow Marine Northumberland Senior Cup next month, while holders Porto visit Tyneside on a date to be confirmed in the semi-finals of the Premier League International Cup.
The full Premier League Cup quarter-final draw is:
Sheffield United or Derby County v Newcastle UnitedLeicester City v Wolverhampton WanderersSwansea City or Bristol City v Brighton & Hove AlbionMiddlesbrough v Aston Villa