Beckford's brace means he is the first Leeds player to score 31 goals in a season since Lee Chapman in 1990-91.
The striker broke the deadlock in the 37th minute, delivering a blistering volley past goalkeeper Willy Gueret after Luciano Becchio headed down Ben Parker's free-kick.
He could have had a second four minutes later had it not been for the vital intervention of left-back Dean Lewington as the promotion-chasing Dons tried to find their composure.
The visitors, who had drawn six of their last eight matches, channelled most of their attacks down the left where Jason Puncheon's pace and trickery tested a Leeds defence in which on-loan Sam Sodje was reassuringly solid until limping off just past the hour.
By then Beckford had settled the game with his second goal, exchanging passes with the impressive Robert Snodgrass before shooting just inside the right-hand post after 54 minutes.
The Leeds marksman should then have completed his hat-trick but blazed wide after being put through by Richard Naylor. Sam Baldock wasted the Dons' best chance, screwing the ball wide from six yards, as Leeds moved to within a point of their visitors and crucially five clear of seventh-placed Tranmere.