Posh goalkeeper Joe Lewis had been on top form, but there was nothing he could do to prevent Beckford's second-half goals. Craig Mackail-Smith grabbed his 19th goal of the season on 80 minutes but another substitute, Jonny Howson, sealed Leeds' win in injury time.
Looking to avenge a 2-0 defeat at London Road in October, Leeds had Andrew Hughes at right-back in place of injured skipper Frazer Richardson and brought in winger Andy Robinson.
Winger Chris Whelpdale was passed fit for Peterborough , who drew 0-0 with MK Dons in midweek.
Leeds fans were quick to bellow insults at Peterborough manager Darren Ferguson, son of Manchester United boss Sir Alex Ferguson, and the high-flying visitors almost fell behind on 12 minutes when Rui Marques had a close-in shot blacked on the line by Craig Morgan.
Both goalkeepers excelled with first-half saves, Leeds' Casper Ankergren twice making spectacular stops from George Boyd and Lewis also showing lightning reactions to save shots from Robinson.
Boyd was a constant threat on Posh's left flank, the former Stevenage Borough man creating space for himself and his colleagues. From one of his many incisive passes, Aaron McLean prodded the ball against a post but a linesman's flag was raised for offside.
The same combination almost forced a breakthrough on 52 minutes, but from Boyd's pass, McLean shot over the bar from 15 yards.
The turning point came when Leeds replaced Luciano Becchio with Beckford in the 58th minute. Four minutes later Beckford was impeded by Morgan and after Lewis failed to hold Bradley Johnson's sizzling free-kick from 20 yards, Beckford pounced to stab home.
Then, on 71 minutes, Lee Trundle played Beckford through the middle and the deadly marksman brilliantly wrong-footed Morgan before beating Lewis with a peerless finish.
Leeds' defence eventually cracked when Boyd headed against the bar and from Richard Naylor's attempted clearance, Mackail-Smith stabbed the ball home from ten yards.
But after fluffing one chance, Howson rammed home a pass from Robert Snodgrass a minute into time added on to ensure a home win.