Leeds' victory on Saturday was their seventh in a row at Elland Road but they were well below their best.
Swindon, who had gone five games without a win in League One, showed few attacking ideas and it was only careless finishing from the home side and goalkeeper Phil Smith's agility which kept the first half goalless.
Smith, recalled in place of Peter Brezovan, made excellent saves to deny Beckford, Robert Snodgrass and Jonny Howson, while Fabian Delph drove wide of an open goal after Smith had brilliantly parried Howson's shot.
Beckford, who ironically had one of his least effective games, had a 64th-minute header cleared off the line by Michael Timlin, but when Snodgrass sent over a cross from the right he got his head to the ball at the far post for the winner.
Beckford was sent off when he followed an earlier booking by strongly disputing an offside decision.