The visitors took the lead inside three minutes as a long ball played over the top by keeper Casper Ankergren found its way to Beckford who volleyed into the net.
The advantage was doubled seven minutes later when Robert Snodgrass fired the ball home after being picked out from a cross from the left by Fabian Delph.
Chester pulled a goal back after 15 minutes through Ryan Lowe, whose shot deflected off defender Rui Marques and looped into the net. But any hope the home side had of turning the game around soon ended as Beckford added his second after 25 minutes with a vicious left-footed strike from 20 yards.
A fourth Leeds goal was added after 31 minutes when Andy Robinson sidefooted home before Beckford completed his hat-trick four minutes later with a superb lob.
Chester grabbed a late consolation with 15 minutes remaining as Lowe fired home from the spot after Damien Mozika had been felled in the box by Fraser Richardson.