Not surprisingly Leeds kept the side which trounced Yeovil 4-0 in midweek, though there was a place on the bench for striker Liam Dickinson after he joined the club on loan from Derby County 24 hours earlier.
Swindon's only change was Phil Smith's return in goal for Slovakian Peter Brezovan and it was Smith's heroics which repeatedly sprang to the Robins' rescue.
Smith swooped low to his left to prevent Beckford from adding to his 28-goal tally as early as the third minute.
Then the agile keeper turned a powerful drive from Robert Snodgrass round a post and after the lively winger had sent an angled effort fizzing just over the bar, Smith was in action again.
This time the former Crawley keeper did well to parry Jonny Howson's shot and Fabian Delph put the rebound wide when he should have scored.
Swindon had little to offer in attack, Sean Morrison's long throws providing the main threat to Leeds' under-worked rearguard until leading scorer Simon Cox stooped to head inches wide of the near post from Owain Tudur-Jones' free-kick from the left.
Beckford had a header cleared off the line by Michael Timlin and a shot well saved by Smith, but persistence finally paid off as he headed home a cross from Snodgrass at the far post.
In injury time Beckford collected his second yellow card for strongly disputing an offside decision.