Everton's Steven Pienaar hit the bar soon after while from Mikel Arteta's smart free-kick Jack Rodwell glanced a second-half header just inches wide.
Teenager Jack Rodwell scored two fine goals and Louis Saha also notched a brace as Everton look set for the group stages of the competition.
City raised the tempo in the second half but after finding no way through, Arteta slotted home a fine second after a superb run from Jack Rodwell.
Fellaini, along with impressive fellow-midfielder Jack Rodwell, was beginning to have a greater influence on proceedings but it was Osman and Pienaar who combined to put Everton ahead.
Ramires hit the home side's post with a shot late on, while Nicolas Anelka was denied by Tim Howard and Jack Rodwell had an effort saved by Cech.
Before Sunday's game at the Reebok Stadium, Everton's squad was depleted by injuries to in-form striker Louis Saha, who scored four goals against Blackpool, and midfielder Jack Rodwell.
Saha had another effort saved before the moment which transformed the mood inside Goodison Park when substitute Jack Rodwell's overhit pass was miscontrolled by Distin, allowing Liedson to break clear.
When fleeting opportunities did fall the home side's way, Fellaini headed over and a Jack Rodwell nod towards goal was diverted above his own goal by Wigan midfielder Ben Watson.
It was Everton's two late substitutes who clinched the three points as Dan Gosling slid in Steven Pienaar's cross with 14 minutes left and Jack Rodwell made it three with a solo effort in the dying seconds.
Everton's only serious response was a header from midfield man Jack Rodwell that flew just wide and Moyes made an alteration the situation demanded at the interval, removing defender Tony Hibbert in favour of striker Jermaine Beckford.
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