Chris Brunt's second-half penalty looked to have secured the Baggies' place in the fourth round draw until a Fellaini equaliser and Neville finish in extra time turned it round
A Chris Brunt penalty on the 56th minute was enough for the visitors looked to have been enough to secure their place in Saturday's draw after Jan Mucha brought down Peter Odemwingie in the hosts' box, after a turgid opening hour. However, an 89th minute strike from Belgian Marouane Fellaini took the game to extra time with the hosts on the brink of elimination.
Then Roy Hodgson, returning to Merseyside for the first time since being sacked at Liverpool, saw his side concede an extra time strike from Phil Neville as Everton marched into the last 16.
Although seeing six changes to the team that beat Wigan late in the game on Saturday, David Moyes’ starting XI still suggested that this is a competition that the Toffees have no intention of taking lightly, even if the majority of their performance suggested anything but.
First starts for Denis Stracqularsi and Royston Drenthe were expected, while the introduction of Ross Barkley from the off left the Blues faithful hoping to see another strong showing from the School of Science’s latest prodigy.
The visitors approach was far more indicative of a side whose priorities lie within the confines of the Premier League. With Jonas Olsson and Peter Odemwingie the only survivors from the Baggies side that lost to Swansea at the Liberty Stadium, there was little question at kick-off that Hodgson had an eye on his side's clash with Fulham at the weekend.
The clash was Hodgson's first trip back to Merseyside since his disastrous spell in charge of Liverpool and if his return was made hesitantly, then it was matched by his side's opening, with Albion sitting back and allowing the home side the majority of possession in the early exchanges.
Both sides were looking for a catalyst during a turgid opening and it was Somen Tchoyi that nearly sparked the Baggies into life after quarter of an hour, as he weaved in and out of the Everton defence before smashing an effort high into the Gwladys Street End.
It was a strike indicative of an opening half hour low on quality but it was the Baggies that continued to have the better chances, with Peter Odemwingie capitalising on a loose pass from Phil Jagielka in the Everton half before smashing it wide.
50-50 | Odemwingie and Baines go in for a challenge as both sides fail to impress
If disinterest is the potential deathnail for the League Cup, then the first half displayed such insipidity from both sides that it offered no hope of proving the antidote for the supposedly ailing competition. With no comparison to the verve and electricity generated in the Premier League, a conductor was desperately required under the Goodison floodlights in the second 45 minutes.
It was to come in the form of Seamus Coleman who broke down the right early only to provide the Toffees a threat and his side responded accordingly, with the best chance of the game falling to Stracqualursi after Fellaini's knock down. The new boy hit his effort straight at Fulop but for the home side, it was at least a statement of intent.
The Argentine's profligacy was to be punished soon after as Odemwingie latched on to a fine Dorrans through ball before being brought down by Mucha, who had rushed out and found himself caught in no man's land. Possibly fortunate to escape with only a yellow card, the Slovak could do nothing to stop Brunt's resulting penalty, as the visitors snatched the opener from 12 yards.
Moyes responded by dispatching Tim Cahill and Apostolos Vellios off the bench but neither inspired with any real intent, as the Blues looked blunt going forward with the final 20 minutes approaching.
Chances for Vellios and Baines soon passed and Everton would be made to wait until the brink of stoppage time until an equaliser was provided through Fellaini, with the Belgium international smashing a superb left-footed effort past Fulop from all of 20 yards.
It was arguably the game's first real piece of quality but despite having lumbered for 90 minutes, the hosts came alive thereafter, pounding the visitors area until salvation arrived for Hodgson's men in the shape of the whistle signalling the end of normal time.
A less-spirited team would've folded after such a late equaliser but Hodgson's men rallied and could've taken the lead for a second time through Tchoyi, only for the powerful forward to blaze over from a matter of yards.
But then their spirit was all but broken by the Blues, as Neville, still ploughing up the right flank at 34 years old, got himself a rare goal not short on quality after picking up the ball from Drenthe.
The Albion boss may have got revenge over Liverpool with his side's 2-1 win over Kenny Dalglish's side last April but after a defeat as cruel as this, demons will no doubt remain. Merseyside, albeit the blue half, still haunts Roy Hodgson.