George Boateng set the tone for a dire night for the Reds, clipping into his own net following a goalmouth scramble, before Leicester took control of their own destiny through the mercurial Beckford.
The summer signing from Everton headed his first goal in 11 outings for the Foxes before completing his hat-trick with two close-range finishes after the break.
Forest were not without their fair share of chances in the opening half, and Robbie Findley's poor miss with the goal gaping summed up a dreadful evening for Steve Cotterill's side.
The victory represents a morale-boosting second home win in six attempts for Nigel Pearson's side, but heaps further pressure on relegation-threatened Forest, who sit 22nd in the Championship.
They were dominated by their East Midlands rivals from the first whistle.
Neil Danns saw his tricky half-volley punched clear by Lee Camp, and Beckford turned his man to allow Richie Wellens to stab wide within minutes.
The early cut and thrust had the Forest defence in pieces after only seven minutes.
Beckford's innocuous corner was weakly repelled to Danns on the left, the former Crystal Palace man pumping a dangerous ball across to Paul Gallagher, who might have tried to claim the goal as a confused Boateng stubbed through his own net.
Two minutes later David Nugent swivelled to drive the ball across the goal-line but into gloves of Camp.
Kasper Schmeichel nearly handed Forest a way back into the match as he flapped wildly at a tame cross from the left. The prone Dane looked on in horror as the ball spilled to Findley, who inexplicably lifted the ball over from two yards with an open goal to aim at.
But Leicester continued to carve open the lost Reds defence, Gallagher afforded time and space to fizz a firm shot just over Camp's bar.
Cotterill's charges spurned a second gilt-edged chance with 28 minutes played. Findley found a gap behind Paul Konchesky to supply the onrushing Paul Anderson, who hacked wildly at the ball under pressure from Sean St Ledger.
On the half-hour mark, the Foxes made it two.
Lee Peltier, set free by a visionary pass from Wellens, arrived in a good position to shoot from the edge of the box. Camp stretched high but could only propel the ball as far as Beckford, who sprang up to head home off the underside of the bar.
A rare period of pressure preceded a welcome half-time interval for the visitors, culminating in Findley's free header being nudged into the grateful palms of Schmeichel.
But the second half belonged solely to Beckford.
With 50 minutes on the clock, the ex-Leeds striker eluded his marker to latch onto an unclaimed long ball and send a deft flick over the flailing Camp.
The hat-trick was complete less than seven minutes later. Beckford was denied in a one-on-one with Camp but snatched the loose ball to convert before the Reds defenders could react.
Forest were beaten, humbled within an hour by their great rivals, and still the blue shirts poured forward.
Leicester then began to pass in triangles, and Danns tested Camp with a low shot.
Gallagher's late effort failed to hit the target late on, but Leicester were by then already assured of a fourth-round tie against Swindon.