United wasted an early chance when Beckford's ninth-minute shot was cleared off the home line by Joe Skarz.
Ten minutes later Bradley Johnson hammered a 20-yard free-kick against the underside of the Hartlepool bar before it bounced clear.
At the other end, Casper Ankergren did well to save Joel Porter's header and to smother at the feet of Daniel Nardiello.
Leeds went close again after 28 minutes when defender Sam Sodje's header hit the angle of the home bar and post.
Leeds took the crucial lead on the hour-mark when Beckford turned on Ben Parker's cross in the area before slotting a low shot into the bottom corner.