Beckford had opened the scoring in the 13th minute, but Exeter levelled through Alex Russell 17 minutes from time, after former Leeds trainee Barry Corr had been sent off for the visitors. But Beckford popped up late on to net his second and deny the battling Grecians a point.
Beckford had missed two good chances before he coolly opened the scoring when left-back Ben Parker set off on a run through the middle and slipped a through pass to Beckford who cleverly tucked a low shot into the bottom left corner.
Beckford headed against the crossbar from Jason Crowe's 25th-minute cross, while Corr spurned Exeter's only first-half chance at the other end when he blazed wide of the near post.
He was then sent off for an elbow on Michael Doyle in the 59th minute, but Russell levelled when he tapped a free-kick to Ryan Harley and then struck the return pass beyond Shane Higgs.
However, Robert Snodgrass and Crowe combined to set up Beckford's late winner.