Leeds, who had leaked ten goals in their last four matches, welcomed back experienced defender Paul Telfer who replaced Angola international Rui Marques.
There were also places for Ben Parker and Jonny Howson, while Orient brought in Aiden Palmer, Alton Thelwell and John Melligan.
Beckford should have given Leeds a tenth-minute lead from the penalty spot after referee Colin Webster ruled that Melligan handled Parker's driven cross, but the striker lost his footing as he spooned the ball high over the bar.
Despite their position in the relegation zone, Orient played some neat football, Paul Terry and Adam Boyd firing over from good positions before Leeds keeper David Lucas spread himself to make a desperate near-post save from Dean Morgan.
However, Morgan wasn't to be denied and on 36 minutes he beat Lucas with a fierce long distance shot into the bottom left corner.
Leeds were level two minutes later with an own-goal by Stephen Purches.
Beckford's cross was controlled by Luciano Becchio and in trying the clear the ball, skipper Purches lifted it over keeper Glenn Morris and in off the bar.
Leeds had been on the back foot for long spells but they took the lead in first-half injury-time when Frazer Richardson sent over a perfect cross for Becchio to score with a close-range climbing header.
Beckford volleyed against the bar, stabbed the ball wide of an open goal, headed against a post twice and brought a flying save from Morris in the second half.
But Lucas had to make a brave blocking save to prevent an equaliser from Jason Demetriou.