Colombian striker Rodallega found the bottom corner after 18 minutes with a right-footed effort after good work from Karagounis found Jon Arne Riise, who crossed for his teammate to tap in.
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With five minutes of the half remaining, Morgan pounced inside the area to level things up for the Championship side after Stekelenburg could only parry Anthony Knockaert's free kick.
The Foxes then took the lead on the stroke of half time when Wood benefited from a fortunate bounce to slide the ball home after some slack defending from Fulham when trying to clear their lines.
However, their two goal lead was immediately reduced when Rodallega's header gave him his second goal of the evening two minutes later.
Karagounis looked to have taken the game to an extra 30 minutes after a powerful freekick after 87 minutes, but Dyer pounced even later to give his side the victory, and send them through to Wednesday night's draw.