Jermaine Beckford was the most wasteful, missing chances before and after the break. Beckford again went close in the dying minutes but Everton held on for the point.
Steve Stone's comeback also gathered pace last night after the winger came through 70 minutes.
Stone seemed untroubled by the Achilles problem which has haunted him since his free transfer from Portsmouth last summer, and he was eventually withdrawn with 20 minutes remaining after staking his claim for a first-team opportunity.
Kevin Blackwell said: "Steve's done what we wanted him to do, and last night was another steady step for him. Over the next couple of first-team games we'll look at getting him involved at some point.
"He's a first-class player with a first-class attitude, and that's why I signed him."
Stone's solid performance stood out last night during a game which Leeds dominated but failed to settle. United took the opportunity to field several first-team squad members.
Joel Griffiths, who completed 90 minutes, went close midway through the first half when his shot was saved by Everton keeper Turner, but opposite number Mark Wilberforce was called upon twice to stop efforts from Victor Anichebe.
Gylfi Einarsson looked to have opened the scoring with 19 minutes remaining but his shot rebounded off the crossbar, and United were unable to break the deadlock after using Simon Walton as a makeshift striker during the closing stages.
United: Wilberforce, Harding, Madden, Hird, Walton, Bayly (Howson), Stone (Parker 70), Einarsson, Pugh (Keogh 70), Beckford, Griffiths.
Everton: Turner, S Wright, Molyneux, Boyle, Kissock, Hughes, Harris, Hopkins, Phelan, Anichebe, Vidarsson.
Subs: Ruddy, Dennehy, Morrison, Harpur, Irving.