Phil Neville scored the winning spot-kick on Saturday after Nicolas Anelka and Ashley Cole both missed for the Blues, whose quest for a historic hat-trick of cup wins now lies in tatters.
Chelsea wasted a hatful of chances to win the match in normal time and Frank Lampard finally gave them a 104th-minute lead but Leighton Baines levelled in the 119th minute with a brilliant free-kick.
The full-back missed Everton's first penalty in the shootout but it did not matter as the Toffees set up a last-16 tie at home to Reading, due to be played at Goodison Park on Tuesday, March 1.
Chelsea, who had not scored in their previous two games, were fortunate still to be in the competition after Saloman Kalou netted a late equaliser in last month's original tie.
They had not beaten Everton since the 2009 final and had won only one of their previous six home meetings with the Toffees.
With visiting boss David Moyes having branded Sunday's defeat at Bolton one of the worst performances of his near nine-year reign, the Merseysiders also had a point to prove.
The same could be said for Didier Drogba, who was controversially axed in favour of Fernando Torres for Monday night's goalless draw at Fulham.
But the Ivorian looked far from sharp early on and needed several minutes of touchline treatment after an innocuous Phil Jagielka challenge in the 14th minute.
Chelsea should have taken a 20th-minute lead when the post saved Jagielka from glancing in an own goal from Lampard's free-kick and John Terry skied the close-range rebound.
Blocks from Sylvain Distin and Howard then prevented Lampard and Florent Malouda putting the home side ahead after excellent work from the recalled Kalou down the right.
Drogba then held off Jagielka and produced a brilliant through-ball for Malouda, whose clipped finish was saved by the feet of Howard.
Malouda picked up the game's first yellow card for clattering into Seamus Coleman and Ramires followed in controversial fashion on the stroke of half-time after being adjudged to have dived in the box under a challenge from Howard, when there appeared to have been at least some contact.
Ancelotti withdrew John Obi Mikel for the axed Michael Essien during the interval.
Baines and Coleman were booked for fouls on Ramires and Branislav Ivanovic before bad marking saw Lampard send a free header off target from Drogba's free-kick.
The recalled Leon Osman almost made him pay by glancing a Baines free-kick too close to Petr Cech before Distin was carded for scything down Ivanovic and Chelsea missed two gilt-edged chances to break the deadlock 25 minutes from time.
Lampard made a hash of a close-range finish from Ramires' cross, allowing Howard to block, with Ivanovic then seeing another point-blank effort thwarted by men on the line.
Ramires' grasscutter drifted wide before Diniyar Bilyaletdinov replaced Tim Cahill for the final 20 minutes.
Buoyed by Chelsea's wastefulness, Everton went close themselves when a good move ended with Coleman heading too close to Cech.
The hosts became increasingly desperate and Lampard flashed wide from 25 yards before Everton withdrew the disppointing Beckford for Victor Anichebe.
The substitute immediately went close to connecting with Bilyaletdinov's cross but Chelsea missed their latest golden chance when Lampard chipped over the bar from Malouda's lay-off.
With just a minute remaining, Marouane Fellaini thought he had dumped the holders out when he netted after Baines' right-foot strike had been fumbled by Cech, but the midfielder had already been flagged offside.
Chelsea finally threw on Anelka at the start of extra-time, Malouda making way, with Everton soon following by introducing John Heitinga for Osman.
Drogba chipped just over and Lampard screwed a low left-footer wide before finally putting the home side ahead thanks to great work from Anelka.
The substitute got himself free down the right and his cross was miscontrolled by Drogba but fell perfectly for Lampard to rifle home from 10 yards.
Chelsea withdrew Kalou for the final 10 minutes, with Yury Zhirkov making his first appearance for more than three months.
Mikel Arteta was carded for felling Essien as Everton tried to level and they did so with seconds remaining thanks to a brilliant Baines free-kick.
Ivanovic fouled Jagielka 25 yards out and Baines stepped up to curl a wondeful strike into the top corner.
Howard spilt Essien's 25-yarder in stoppage-time but regathered to take the tie to penalties.
Chelsea had lost six of their previous seven shootouts but held an immediate advantage when Cech saved from Baines after Lampard had scored.
However, Anelka foolishly tried to chip Howard with the hosts' third penalty and it stayed level until Ashley Cole ballooned over the bar with their fifth.
Everton captain Neville stepped up and found the top corner to seal an amazing win.