The country's leading marksman's chance from the spot arrived after only 10 minutes but he slipped during his run-up, hoisting the ball high over the bar after JJ Melligan was adjudged to have handled Ben Parker's cross from the left.
The miss looked like proving costly when Orient took the lead after 36 minutes, Dean Morgan gaining possession 35 yards out before darting forward and striking the ball into the bottom left corner from outside the penalty box.
But Leeds struck back two minutes later when Beckford's right-wing cross was controlled in the goalmouth by Luciano Becchio, before Orient captain Stephen Purches swung his foot at the ball which looped over keeper Glenn Morris and into the net off the underside of the bar.
Surprisingly, Orient had been the more incisive team but Becchio put Leeds ahead in first half injury-time with a powerful six yard header from Frazer Richardson's measured centre.
Beckford hit the bar once and a post twice in the second half and was the victim of a flying save from Morris but Leeds keeper David Lucas needed a blocking save to deny Jason Demetriou an equaliser.