Wayne Routledge scored an injury-time winner for Fulham to cap a stunning comeback and book a home tie against Stoke in the FA Cup fourth round.
Montella inspired the Fulham comeback at Craven Cottage
Matt Fryatt shot across Tony Warner to put Leicester ahead before Brian McBride equalised from close range.
Gareth McAuley headed his first goal for Leicester and James Wesolowski struck from 18 yards to make it 3-1.
But Fulham substitute Vincenzo Montella scored twice from close range before Routledge's mazy run and finish.
It was a thrilling cup tie - that Leicester seemed to have under control when substitute Wesolowski struck shortly after the break to extend his team's lead.
But Montella - making his home debut for Fulham following his loan move from Roma - showed a real poacher's instinct with his brace before Routledge scored his first goal for the club to snatch victory.
The opening phase of the game belonged to Leicester - with the Championship side playing with real purpose in front of prospective new owner Milan Mandaric.
Fryatt opened the scoring with the first clear chance of the match on 13 minutes after collecting a long pass from Stephen Hughes.
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He then forced a decent save from Tony Warner after cutting in from the right and shooting across goal.
Gareth Williams almost embarrassed the Fulham keeper with a long-range strike that swerved viciously, forcing Warner to make a late adjustment and bundle the ball out for a corner.
Alan Maybury tried his luck from distance as Leicester continued to press - before McBride equalised.
The ball fell invitingly to the American after Patrick Kisnorbo failed to head clear Tomasz Radzinski's shot and McBride slotted home with a low strike from five yards.
Fulham warmed to the task after scoring, but the last meaningful action of the half belonged to the visiting team.
Leicester defender McAuley made no mistake with a far-post header, burying Levi Porter's free-kick.
And just after the restart Leicester substitute Wesolowski - who had replaced the injured Danny Tiatto - extended his team's advantage with a thunderous strike that flew into the top corner.
Any thoughts that the goal might have put the tie beyond Fulham proved unfounded - as Fulham substitute Montella, on for McBride, inspired a scintillating comeback with two goals in eight minutes.
He crashed a shot against the post before slotting the ball into an open goal seconds later after a low cross across goal from skipper Michael Brown.
The Italian then climbed highest to head home at the far post after a Leicester player only managed to flick a Routledge corner in his direction.
It was a breathtaking passage of play from a team who had seemed devoid of inspiration for much of the tie.
Franck Queudrue looked certain to score a fourth for Fulham but his header struck Maybury close to the Leicester goalline and flew out for a corner before Carlos Bocanegra saw a header saved by Paul Henderson.
Leicester seemed to have held on for extra-time but were undone by Routledge's jinking run.
He flicked the ball past two Leicester players, shook off another and, as the referee played advantage for a foul, advanced into the penalty box and placed the ball beyond Henderson.
Fulham manager Chris Coleman:
"The scoreline was great, lots of goals and excitement, but the first-half performance was woeful. We deserved to go in 2-1 down.
"But we do not know when we are beaten and the second-half was very good.
"We gave ourselves a mountain to climb but we managed to do it. We have not been to the fourth round for a while."
Leicester boss Rob Kelly:
"The bottom line is that we lost. We came to an established Premiership team and scored three goals but did not win.
"But we will take the positives and try to take that level of performance into the Championship.
"We are a young side and we showed that we can compete."
Fulham: Warner, Rosenior, Christanval, Bocanegra, Queudrue, Routledge, Volz, Brown, Radzinski, Helguson, McBride (Montella 46).
Subs Not Used: Lastuvka, Runstrom, Zakuani, Omozusi.
Goals: McBride 35, Montella 51, 60, Routledge 90.
Leicester: Henderson, McAuley (Stearman 81), Kisnorbo, McCarthy, Maybury, Hughes (Cadamarteri 82), Williams, Tiatto (Wesolowski 38), Porter, Hume, Fryatt.
Subs Not Used: Logan, Johansson.
Goals: Fryatt 13, McAuley 45, Wesolowski 47.
Ref: A Wiley (Staffordshire)