City were also denied a clear penalty in the 86th minute when referee Lee Probert awarded the visitors a free-kick for a hand-ball by Marouane Fellaini that occured inside the 18-yard box.
Beaten semi-finalists in last season's competition, Everton were without the injured Marouane Fellaini but made a purposeful start and took the lead in the 15th minute.
Leon Osman and Steven Naismith scored the goals as Everton fought back from two goals down to draw 2-2 in an engaging Merseyside derby with Liverpool on Sunday.
Leon Osman's deflected shot and Victor Anichebe's second-half strike gave Everton a 2-0 win at home to West Bromwich Albion in the Premier League on Saturday.
Everton boss David Moyes has backed Tim Cahill to rediscover his goal-scoring form despite a year-long drought which contributed to the Australian being dropped from the side.
Leon Osman's late equaliser earned Everton a 1-1 draw at home to Norwich City on Saturday and prevented David Moyes' side from slipping to a third consecutive defeat.
Leon Osman continued Everton's jinx over Manchester City as his second-half winner sealed a 2-1 comeback victory that dented the visitors' push for a Champions League place.