Leeds saw off the challenge of Darlington in the northern section, second round to secure their progress in the Johnstone's Paint Trophy.
Central defender Paul Huntington grabbed the winner for the League One side when he headed in a Jonathan Howson corner early in the second half.
Mark de Vries then struck the woodwork for the visitors after meeting a Wayne Andrews cross.
Matthew Green hit the post with a shot at the other end but Leeds held out.
Darlington: Oakes, Purdie, Foster, White, Ryan, Joachim,
Keltie, Ravenhill (Cummins 78), Barrua (Wainwright 56), Blundell, Wright (Green 24).
Subs Not Used: Miller, Brooks-Meade.
Booked: Blundell, Keltie, White.
Leeds: Lucas, Richardson, Huntington, Heath (Clapham 46),
Parker, Weston, Howson, Thompson (Da Costa 57), Westlake,
Andrews (Ameobi 78), De Vries.
Subs Not Used: Martin, Bayly.
Booked: Andrews, Thompson.
Goals: Huntington 48.
Ref: Michael Jones (Cheshire).