The faulty Foxes fell to a 3-0 deficit in nine shocking first-half minutes with Philip Zinckernagel, Brennan Johnson and Joe Worrall netting for Steve Cooper's men.
Kelechi Iheanacho pulled one back for Leicester just before the break as he capitalised on some poor decision-making from Forest goalkeeper Brice Samba.
However, the tie was sealed on the hour mark. as Djed Spence poked the ball home.
It was a fantastic performance all round from Forest but on loan Middlesbrough man Spence was given special praised by Trevor Sinclair, who labelled the rumoured Arsenal and Tottenham target a 'Premier League player.'
Forest will face fellow Championship side Huddersfield Town at home in the fifth round. Ties will take place on March 2.
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