Crystal Palace beat Hull City in their Fifth Round tie on Thursday night.
The Magpies booked their place in the last eight with a resounding 3-0 success at West Ham.
United also avoided dark horses Blackburn Rovers, who beat favourites Man United 3-0.
FA YOUTH CUP DRAW
Watford v Chelsea
Blackburn Rovers v Everton
Newcastle v Crystal Palace
Fulham v Aston Villa/Millwall