The Toffees had seen their hopes of progressing from Group I diminish after back-to-back defeats at the hands of Benfica last month.
However, Diniyar Bilyaletdinov's sixth-minute strike secured only a second win in 12 games for David Moyes' side and ensured they will be playing in Europe after Christmas and not the Greek side.
Everton were again without a raft of first-team players with the likes of Phil Neville, Mikel Arteta, Jack Rodwell, Leon Osman and Victor Anichebe all missing through injury.
It got worse for Moyes when Dan Gosling had to be replaced on 10 minutes by Jose Baxter after falling awkwardly on a sodden pitch made almost unplayable following a pre-match downpour.
Sylvain Distin limped off on 18 minutes with a recurrence of a hamstring problem which had placed his participation in jeopardy and another youngster in Shane Duffy entered the fray.
But in-between, the Toffees took the lead when Bilyaletdinov crashed home a shot from 10 yards to set them on their way.
Brazilian striker Jo did well to hold the ball up and lay it off to the Russia midfielder, who slammed an unstoppable shot into the roof of the net.
The home side were sluggish in comparison and the closest they came in the first half was a shot from distance on 30 minutes from Panagiotis Tachtsidis which was straight at goalkeeper Tim Howard.
Everton seemed content to sit on their slender advantage in the second period and that allowed AEK to offer more of an attacking threat.
The arrival of big-guns Martin Ignacio Scocco and Ismael Blanco off the bench seemed to pep up the home side and they came close to restoring parity when Youssouf Hersi's angled shot was inches away from nestling inside the far post.
Home keeper Sebastian Saja kept his side in the game with a superb fingertip save to deny the rampaging Jo on the hour-mark when he looked set to find the bottom corner after a strong run down the inside left channel.
That was Jo's last taste of the action as he picked up a knee injury and had to be replaced by Yakubu.
Stand-in right-back Seamus Coleman came to Everton's rescue in the 75th minute when he cleared a shot from Gustavo Manduca off his own line with Howard beaten.
AEK dominated possession in the closing minutes but failed to find a way through Everton's hard-working defence on a pitch which resembled a ploughed field.
The Merseyside club now know that they have qualified for the next stage of the competition with one group match still remaining.