Leicester began on the front foot and were almost ahead when Guy Moussi cleared Neil Danns's volley off the line.
They soon took the lead when George Boateng turned a Lloyd Dyer cross into his own net.
Beckford picked up his first on the half-hour mark by flinging himself at a loose ball after Lee Camp failed to hold on to a cross.
The former Everton man made it three by capitalising on Moussi's error and lobbing Camp before securing his hattrick by reacting first to a rebound.
Steve Cotterill's Forest have now only scored in one game of the past 11 while the Foxes will host Paulo Di Canio's Swindon Town in the fourth round.