Luke Murphy found himself in a good position to put the home side ahead with 11 minutes on the clock, only for Craig Cathcart to make a vital challenge to deny the midfielder.
Just 10 minutes later Murphy was not to be denied, however, scoring from close range following some good work from Ross McCormack to make it 1-0.
Leeds turned up the pressure in search of a second goal and nearly found it thanks to McCormack, with the striker glancing Scott Wootton's cross just wide of the target.
Murphy added his second of the afternoon in the second half when he converted Noel Hunt's flick-on, giving the Whites some breathing space.
McCormack came close to adding a third for Leeds, only to be denied by Matt Gilks after a cross into the box from Murphy, although it was not to matter with Bryan McDermott's charges holding out for the three points.