Gylfi Einarsson put United ahead in the 17th minute but the Quakers drew level with a stunning volley from Fabio Digenti 10 minutes after the break.
But Beckford's quickfire brace restored the Whites advantage and Hayden Foxe wrapped up the win late on.
For Leeds' first goal Frazer Richardson linked up well with Danny Senda and from Richardson's cross the home keeper David Stockdale made a mess of catching it and Einarsson tapped the ball home.
The Icelandic international then went off for his troubles as he picked up an injury and was replaced by Sam Hird.
Darlington equalised on 55 minutes when a cross by Kalusivikako Ngoma was volleyed home by Digenti. A brilliant goal worthy of any game.
Beckford put Leeds ahead on 70 minutes with a great finish after bringing the ball under control and then he scored his second 5 minutes later when a shot by Darlington old boy Robbie Blake was only pushed away by Stockdale straight to Beckford.
Trialist Foxe wrapped things up with 2 minutes left after heading Blake's free-kick into the far corner.
Darlington (4-4-2): Stockdale, Collins (Janes 64), Tavago, Martis (Taylor 12), James (Duke h/t), Dejente (Mulligan 57), Keltie (Ngoma h/t), Rowson (Cummins h/t), Carlos Logan (Wainwright h/t), Giallanza (Conlon h/t) (Richard Logan 71), Smith (Johnson h/t)
The Whites (4-4-2): Bennett, Richardson, Foxe, Gregan, Harding, Senda (Madden 72), Bailey, Einarsson (Hird 18), Morris (Gardiner 65), Beckford, Blake.
Subs Unused: Gardiner, Howson, Camfield.
Bookings: Collins 40 mins
But Beckford's quickfire brace restored the Whites advantage and Hayden Foxe wrapped up the win late on.
For Leeds' first goal Frazer Richardson linked up well with Danny Senda and from Richardson's cross the home keeper David Stockdale made a mess of catching it and Einarsson tapped the ball home.
The Icelandic international then went off for his troubles as he picked up an injury and was replaced by Sam Hird.
Darlington equalised on 55 minutes when a cross by Kalusivikako Ngoma was volleyed home by Digenti. A brilliant goal worthy of any game.
Beckford put Leeds ahead on 70 minutes with a great finish after bringing the ball under control and then he scored his second 5 minutes later when a shot by Darlington old boy Robbie Blake was only pushed away by Stockdale straight to Beckford.
Trialist Foxe wrapped things up with 2 minutes left after heading Blake's free-kick into the far corner.
Darlington (4-4-2): Stockdale, Collins (Janes 64), Tavago, Martis (Taylor 12), James (Duke h/t), Dejente (Mulligan 57), Keltie (Ngoma h/t), Rowson (Cummins h/t), Carlos Logan (Wainwright h/t), Giallanza (Conlon h/t) (Richard Logan 71), Smith (Johnson h/t)
The Whites (4-4-2): Bennett, Richardson, Foxe, Gregan, Harding, Senda (Madden 72), Bailey, Einarsson (Hird 18), Morris (Gardiner 65), Beckford, Blake.
Subs Unused: Gardiner, Howson, Camfield.
Bookings: Collins 40 mins