West Ham and Everton both selected strong sides, an indication that the stakes are now higher at the business end of the competition.
There was a scare when the floodlights went out briefly 15 minutes before kick-off but once the action got under way Everton were quickly into their stride.
But West Ham turned the tables and went ahead after only 12 minutes when Neill's long pass caught out the Everton defence, allowing Cole to beat Everton keeper Tim Howard to the ball and slide in the finish.
Everton remained composed and almost equalised when keeper Green was lucky to turn Steven Pienaar's cross away from Tim Cahill.
But the pressure told and Everton were level five minutes before the interval when Pienaar played in Osman neatly and he curled a deflected shot past Green for a deserved equaliser.
The second half began in ragged fashion, with Pienaar infuriating the Upton Park crowd with his theatrics, throwing himself to the ground unnecessarily on several occasions.
In a tense cup tie, both defences were strong and on top as it failed to match the quality of the opening 45 minutes.
Yakubu's control let him down as Everton created a chance with 15 minutes to go, then Freddie Ljungberg's hesitation cost West Ham an opening after Mikel Arteta's casual back-heel left his side open to a counter-attack.
It looked like only a mistake would divide the sides - and so it proved as Everton snatched the winner with two minutes left.
Phil Jagielka's long ball looked harmless until keeper Green and Gabbidon got in a tangle and the ball rolled through for Yakubu to complete an easy finish.
West Ham manager Alan Curbishley: "We started ever so well, got the goal, and unfortunately Everton took over after that. They have a very established side.
"The referee had a torrid game but the big difference was they were a bit sharper and more physical.
"It was drifting into extra time and then they got their goal. Danny and Greeny got caught in between. Greeny was going to get it, Danny headed it. It was a mistake."
Everton manager David Moyes: "The way we came back after they scored was terrific. The first-half might have been as good as we have played in my time here.
"It is an important result and we are delighted with it, but we have not won anything yet.
"We are going to really enjoy this result and the players and fans deserve it, but you don't get anything for winning a quarter-final."
West Ham: Green, Neill, McCartney, Gabbidon, Upson, Ljungberg, Parker, Mullins, Boa Morte (Reid 67), Cole (Pantsil 76), Ashton. Subs Not Used: Wright, Noble, Spector.
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