With the game tied at 3-3 and heading for extra-time, Routledge, on loan from Tottenham, weaved into the box to stab home the winner.
The Premiership side looked to be heading out of the FA Cup when Leicester sailed into a 3-1 lead.
But two goals from Vincenzo Montella and Routledge's superb solo effort in stoppage time set up a home tie with Stoke in round four.
Leicester certainly didn't play like a team languishing 18th in the Championship table. Striker Matt Fryatt was clearly a useful outlet in attack and, after a bright start, it came as no surprise when the Foxes took the lead.
Stephen Hughes provided the assist with a precise chip over the top and Fryatt ran clear of Carlos Bocanegra to coolly slot the visitors into a 13th minute lead with his first goal since November.
Moments later, the England Under-20 international came close to adding another when he beat the Fulham offside trap again. Fryatt picked up Danny Tiatto's pass, but this time his left-foot shot from the edge of the area was well-saved by Tony Warner.
Gareth Williams then tested Warner from distance before Fulham finally got their act together. Tomasz Radzinski made good strides down the left to cross for Brian McBride to fire in from close range.
But Leicester were clearly not overawed by their Premiership opponents and went into the break in front following Levi Porter's free-kick.
The youngster's inswinger from the right caused havoc at the far post and Gareth McAuley forced the ball home.
And, within seconds of the restart, a shock looked to be on the cards when Leicester doubled their advantage. Fryatt again was involved as he teed up sub James Wesolowski to fire in his first goal in Leicester colours from the edge of the area.
But Fulham had a trump card up their sleeve in the form of former Italy international Montella.
The striker, on loan from AS Roma, had come on at half-time for McBride and it took him just ten minutes to prove his worth and get Fulham back on level terms.
First the 32-year-old found space to fire a left-foot shot against the post before Michael Brown crossed for him to tap in on 52 minutes.
And just eight minutes later, the man dubbed in Italy 'The Little Airplane' rose highest at the far post to head in Routledge's deflected corner for Fulham's leveller.
It certainly buoyed the home side and Leicester had defender Patrick Kisnorbo to thank on 67 minutes for keeping them level. Franck Queudrue's header from another Routledge corner looked goalbound until Kisnorbo stuck out his head to divert the ball over the bar.
But with extra-time looming, Leicester's brave cup run came to an end thanks to a bit of magic from Routledge in stoppage-time.
The winger beat four challenges before poking him from close range to send the Cottagers into the next round.