United's second-string qualified for the last-eight stage as Group C winners after recording eight points, defeating the likes of FC Halifax Town, Tamworth and Gateshead.
Their reward is an away tie against Altrincham, who currently occupy seventh spot in the English fifth tier, and the cup contest must be played by Wednesday, 5th March.
A date and kick-off time will be confirmed on newcastleunited.com in due course.
National League Cup quarter-final draw
Aldershot Town vs Leeds United Under-21s
Altrincham vs Newcastle United Under-21s
Braintree Town vs Brighton & Hove Albion Under-21s
Sutton United vs Manchester United Under-21s