Leeds were reduced to ten men in only the ninth minute when Alan Sheehan was shown the red card by referee Darren Deadman after a poor challenge on Swindon midfielder Jon-Paul McGovern.
Swindon could not capitalise on their one-man advantage and it was Leeds that went on to grab their third straight win with an emphatic victory.
Leeds took the lead in the 22nd minute with Beckford grabbing his ninth goal of the season.
Frazer Richardson crossed from the right and Swindon defender Jerel Ifil somehow let the ball slip under his feet and Beckford nipped in to lashed a shot home into the top right-hand corner of the net.
Swindon undeservedly equalised just before the break through top scorer Simon Cox. Midfielder Anthony McNamee put in a delightful cross from the left-hand side and Cox volleyed home right-footed into the top right-hand corner for his sixth goal of the season.
Leeds regained the lead in farcical fashion five minutes after the restart. Midfielder Neil Kilkenny hit a 30-yard free-kick that Phil Smith in the Swindon goal should have saved easily.
But Smith somehow managed to bundle the ball over his own line to give Leeds the lead. Swindon should have equalised within 30 seconds but Leeds goalkeeper David Lucas pulled off a magnificent save.
McNamee crossed from the left and Cox met the header six yards from goal but Lucas dived to his right superbly to palm the ball away for a corner.
Swindon renewed their efforts but Leeds always looked the more likely to score and again it was that man Beckford who delivered.
Eighteen-year-old midfielder Fabian Delph beat two players before threading a superb throughball to Beckford.
The big striker made no mistake as he ran into the area and fired home right-footed into the bottom left-hand corner to cap a fine display.