Ayegbeni Yakubu pounced on a defensive howler two minutes from time to book Everton a place in their first Carling Cup semi-final since 1988.
With the match seemingly heading for extra-time after a desperate second half Danny Gabbidon headed the ball past his own goalkeeper allowing the Nigerian to poke home the winner.
It was Everton's only shot of the half but was enough to seal the Toffeemen victory over a West Ham side who started brightly but quickly ran out of ideas.
The Hammers had taken an 11th-minute lead through Carlton Cole but Everton always looked more dangerous in possession during the first half and levelled through Leon Osman.
Everton's win extended their unbeaten run to 11 matches in all competitions and they return to Upton Park to tackle West Ham in the league at the weekend.
Dean Ashton and Freddie Ljungberg returned to the Hammers' starting line-up for the first time since October with Matthew Etherington out injured and Nolberto Solano cup-tied.
Everton made just one change from their 3-0 win over Fulham with Phil Jagielka replacing injured full-back Leighton Baines.
After a brief scare when the lights went out 15 minutes before kick-off, Everton engineered the first chance in a cagey opening.
Lee Carsley's drive was deflected wide and West Ham looked vulnerable as Mikel Arteta delivered two testing corners, the second of which Yakubu headed wide.
Steven Pienaar jinked his way into the West Ham box, beating three defenders who were too concerned about conceding a free-kick to make the tackle, before he was eventually squeezed off the ball.
Ashton just failed to control Hayden Mullins' angled ball towards the edge of the penalty area and the ball just skipped away from Ljungberg stealing in at the far post.
But another angled ball from deep, this time from Lucas Neill, undid the Everton defence as West Ham took the lead after 11 minutes.
Cole escaped his man and beat the on-rushing Tim Howard with one touch before clipping the ball in for his fourth goal of the season.
Green comfortably held a long-range drive from Tim Cahill before Ashton almost slipped Cole in again but the through-ball was just too heavy.
Scott Parker, Ashton and Ljungberg combined well to tee up an effort for Luis Boa Morte on the edge of the Everton area but the Portugal international blazed his strike over the crossbar.
But West Ham could not maintain the momentum as Everton assumed control.
A scuffed clearance from Matthew Upson invited pressure and caused confusion in the West Ham area. Arteta missed his kick and Cahill went down hard under a fair challenge from Upson.
Green tipped Pienaar's low cross away from Cahill but eventually the pressure told.
With Everton camped in and around the West Ham box, Pienaar slid a neat pass to the unmarked Osman who curled his shot past the stranded Green.
It was no less than Everton deserved for a concerted period of pressure.
The second half was desperate with both sides managing just one shot apiece.
Pienaar's theatrics found no favour with referee Mark Halsey after the midfielder let the ball run across him and went down too easily looking for a penalty.
Mullins linked with Ljungberg and charged into the Everton penalty area but nudged the ball just too far ahead and could not get a strike on goal.
West Ham increasingly resorted to long and ineffectual balls towards Cole and Ashton which Yobo and Jagielka dealt with comfortably and their solitary effort of the half was an instinctive volley from Cole which sailed wide.
With extra-time looming, West Ham gifted Everton victory.
Gabiddon allowed Jagielka's long ball to bounce and just as Green was coming out to claim the central defender headed it through the keeper's arms allowing Yakubu to pounce.
