Chelsea looked to be on course for a place in the fifth round when midfielder Frank Lampard fired them 1-0 up in the first half of extra time at Stamford Bridge.
But a sensational last-gasp equaliser from Leighton Baines, the Everton left-back curling in a free-kick from 25 yards, took the game to penalties.
Even then Chelsea appeared to be on course for victory, when Lampard and Didier Drogba buried their opening spot-kicks either side of a miss by Baines to put the Blues 2-0 up in the shoot-out.
Phil Jagielka kept Everton in it with a clinical penalty before Nicolas Anelka saw his effort palmed away by Tim Howard.
Mikel Arteta scored for Everton to level at 2-2 and when Ashley Cole missed with Chelsea's fifth spot-kick -- sending his penalty blazing over the bar -- it was left to Everton skipper Phil Neville to blast home the winner.
Victory was especially sweet for Everton, who face Reading in the next round, as the Merseysiders had been beaten by Chelsea in the 2009 final.
The defeat leaves Chelsea's domestic season in tatters. They trail Premier League leaders Manchester United by 12 points and their only realistic chance of silverware lies in the Champions League.
Chelsea -- who had been held to a 1-1 draw by Everton in the first match at Goodison Park -- had the upper hand throughout normal time but were unable to convert a plethora of scoring opportunities.
Everton keeper Howard made a string of stunning saves as the Londoners laid siege to Everton's goal.
Everton, meanwhile, thought they had grabbed a dramatic winner in the dying minutes, when Marouane Fellaini prodded home the rebound from a long-range effort by Baines.
However the Belgian international's strike was correctly ruled out for offside, albeit by the tightest of margins.
Chelsea's match was the first of four FA Cup matches on Saturday.
In other matches, Premier League side Stoke entertain League One leaders Brighton, from English football's third tier, while Birmingham also face League One opposition in the form of struggling Sheffield Wednesday.
The tie of the day comes in the 1715 GMT kick-off at Old Trafford, where mighty Manchester United face non-league Crawley Town in the fifth round.
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