Robins manager Paul Sturrock lit the fuse by claiming Swindon had the potential to become an even bigger club than Leeds. Beckford answered him by scoring from the penalty spot and after Simon Cox equalised just after the break, Beckford hit Leeds' winner.
Cox also had a goal disallowed in the 83rd minute for handling on the line after Casper Ankergren parried Barry Corr's header.
With Alan Thompson on the bench and Jonathan Douglas injured, Andy Hughes captained Leeds on his return from suspension.
Tresor Kandol claimed the ball had crossed Swindon's goal-line after only five minutes when his header was held by Slovakian keeper Petr Brezovan, but referee Lee Mason and his assistant disagreed.
Teenager Jonny Howson, who went on to show his class in midfield, was almost punished for losing possession in his own penalty box but Billy Paynter's shot was blocked by keeper Ankergren, much to Howson's relief.
It was Swindon's turn for a let-off when Jamie Clapham's low drive struck Lee Peacock on the line and rebounded into Brezovan's hands.
Wise wants England Under-21s manager Stuart Pearce to watch Howson and the youngster did enough to back the argument with some incisive passing and well-timed tackles.
Leeds seized a 31st minute lead from a controversial penalty. Swindon captain Hasney Aljofree was adjudged to have handled David Prutton's cross fractionally inside the box and referee Mason waved aside protests, before Beckford fired the spot-kick into the top left corner.
Swindon caught Leeds cold 56 seconds into the second half when Cox centred for Peacock to drill a ten-yard shot past the unprotected Ankergren.
Beckford added his second in the 56th minute, taking Kandol's lobbed pass through the middle and scoring on the half-volley from 15 yards.