Aiyegbeni Yakubu pounced on a defensive howler two minutes from time to book Everton a place in their first Carling Cup semi-final since 1988.
Everton had to come from behind after West Ham made a bright start and earned an 11th minute through Carlton Cole.
But Leon Osman equalised shortly before the interval and then Yakubu pounced to poke home the winner after a mix-up between Danny Gabiddon and Robert Green.
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