Peterborough's victory against Leicester earlier in the day reduced Leeds ' chances of claiming one of the automatic promotion spots, but they and the Dons were keen to consolidate their places in the play-off zone.
Leeds gave a debut to Sam Sodje and the central defender, who is on-loan from Reading, won approval from the home fans with some vital clearances before departing with an injury in the 61st minute.
The Dons, who had drawn six of their last eight games, were without Stephen Gleeson, who was on international duty with the Republic of Ireland, while Luke Chadwick, Danny Swailes and former Leeds striker Tore Andre Flo were out injured.
With yellow-booted winger Jason Puncheon quickly into his stride, the Dons set a slick pace and almost took an 11th-minute lead. Leeds captain Richard Naylor brought down Puncheon just outside the box and Sam Baldock's free-kick was pushed round a post with difficulty by keeper Casper Ankergren.
Leeds gradually gained more possession and their patience was rewarded in the 37th minute. The Dons failed to clear Ben Parker's free-kick from the left, Luciano Becchio headed into the goalmouth and Beckford's volley was unstoppable.
Four minutes later Beckford was only denied another goal by Dean Lewington's last-ditch intervention but would not be denied, pouncing again in the 54th minute.
Trading passes with the lively Robert Snodgrass, Beckford shot just inside a post for his tenth goal in his last seven appearances.
With steadier finishing he would have completed his hat-trick on the hour but he shot just wide of the far post.
Baldock missed an even easier chance for the Dons, flicking the ball wide from a few yards out.