The Whites are now winless in 10 league outings, remaining 13th in one of the bottom two relegation places of England's academy top tier.
Injuries within Marcelo Bielsa's senior ranks have filtered down to the development squad which has had an impact on results in recent months.
Fresh from Premier League debuts against Arsenal at the weekend Sam Greenwood captained the Whites while Liam McCarron started in defence.
It was a young backline selected by Mark Jackson amid first team injury problems as Lewis Bate headlined the midfield.
Archie Gray - who was named on the bench by Bielsa against the Gunners - was again part of the supporting cast in York.
The Whites and the Toffees played out an even opening period though it was the hosts who had the better of the play.
Goalkeeper Dani van den Heuvel was forced into a brilliant early save by Charlie Whitaker.
The stopper was unsighted as the visiting man unleashed a thunderous shot at goal but he flung himself to his right to keep the effort out.
Jeremiah Mullen cleared a low shot from distance away from danger that appeared on its way into the back of the net before Kris Moore blocked twice on the line in a frantic passage.
Greenwood went close at the opposite end as Bate and Jack Jenkins began to control the midfield to help create chances for Charlie Allen, Max Dean and Sean McGurk though none were able to convert.
Jackson made three changes at the interval with Gray, Stuart McKinstry and Nohan Kenneh entering the game.
It was Leeds who dominated second half possession but the hosts were again saved by their goalkeeper.
Everton broke away through Isaac Price with United committed forward and the midfielder was left one-on-one but van den Heuvel came to the rescue with his legs once again.
Leeds saw McGurk thwarted from distance - ramping up the pressure late on - while McKinstry and Gray both went close.
Dean, who moved inside from the flank, almost ghosted into the box late on but was denied by visiting goalkeeper Harry Tryer as he turned inside the six-yard box.
The draw leaves Leeds without a victory in the PL2 since September, signing off 2021 with a goalless draw.
"I thought we deserved the three points," Jackson told the YEP post-match.
"They had a few clear cut chances but the way we played and tried to implement our style on Everton was really pleasing.
"The fact we got a clean sheet, I've just said to the boys in there [the dressing room] that we can use this as a platform for us to build on.
"We can take a lot of confidence from that. We're happy with the point but on another day it might've been three."
Leeds United U23s: Van den Heuvel, Sutcliffe, Mullen, Moore (Kenneh 45), McCarron, Jenkins, Bate (Gray 45), Allen, Dean, McGurk, Greenwood (McKinstry 45). Subs not used: Klaesson, Carole.
Everton U23s: Tryer, John, Anderson, Welch, Astley, Price, Mills (Kouyate 60), Whitaker (McAlister 60), Cannon, Warrington, Hughes (Hunt 69). Subs not used: Barrett, Campbell.