But Everton stunned Stamford Bridge by equalising at the death through a curling Leighton Baines free-kick.
In a low-key game, the Gunners enjoyed the majority of the possession but Everton created several goal chances.
His long clearance evades Bolton goalie Adam Bogdan but Everton still lose 2-1.
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But Everton were clearly in control and Osman compounded Hull's misery by pouncing after Heitinga's free-kick was not cleared.
Rooney's current deal still has two years left to run, but Everton are anxious to secure his long-term future.
Kuyt scored, but Everton were enraged when Carragher fouled Lescott in front of referee Clattenburg, who waved away their appeals.
But Everton - who had not played since last week's first leg - were a little rusty in the first half.
BBC: Andrew Johnson and Yakubu celebrate Everton's first goal
The hosts enjoyed all the possession, but Everton had the better chances.
Pavel Sultes swept in an equaliser from distance but Everton eased through.
But Everton are past masters when it comes to party-pooping, and for a few minutes they looked likely to call the festivities to an abrupt halt.
But Everton had other ideas and after they were awarded a free-kick on the edge of area, Baines stepped up to curl a delightful set-piece into Cech's top left-hand corner.
But Everton grabbed the initiative towards the interval and should have equalised in first-half stoppage time, right winger Toni Duggan brilliantly weaving her way past three defenders but then shooting well wide of the far post.
But Everton, who were held 1-1 by Carlo Ancelotti's side in the original tie at Goodison Park last month, had not lost to the Blues since the 2009 FA Cup final, and Neville felt the win was merited overall.
But Everton's defence was expertly marshalled by centre-back Phil Jagielka and stood firm against a Birmingham side who failed to create any chances of note, with Alexander Hleb a peripheral figure as he struggled to provide any creativity or guile.
Dimitar Berbatov's 16th-minute goal looked to have paved the way for United to turn up the pressure on pacesetters Chelsea at the top of the table - but Everton had other ideas as they took another major scalp to emphasise their recent improvement.
But the Everton supporter took pride at the performance of his players against the European champions.
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David Weir is an experienced defender Scotland could sorely do with but the Everton man turned his back on his country after being picked out for public criticism from Vogts.
The Super Eagles should have broken the deadlock on 14 minutes when Kalu Uche set up Yakubu up to beat the offside trap but the Everton forward blasted wide with only Rachad Chitou at his mercy.
But the Everton boss revived his interest in Arsenal's 2005 FA Cup winner with defensive cover looking thin at Goodison Park following Neill's surprise exit and Tony Hibbert facing at least a month out with a hernia injury.
The opening exchanges were tight, but it was Everton who created the better opportunities.
Moyes gave Rooney his debut as a 16-year-old at Everton but they fell out when the England striker left for Old Trafford in 2004.
In the first knockout round, Roy Hodgson's men will avoid fellow English clubs Liverpool and Everton but could face the likes of Juventus, Marseille, Ajax, or Benfica.
But it was Everton who dominated in terms of chances, with Joleon Lescott hitting a post and Louis Saha and Tim Cahill wasteful in front of goal.
It is a result that may not suit either side with Villa's faint hopes of Champions League qualification all but extinguished and Everton's drive toward the Europa League spots losing momentum.
But it was Everton who went closest to a winner when Yakubu found himself free in the box, only to send his drive across goal and away for a goal kick.
Ferguson threw on Michael Owen but it was Everton who finished stronger and Rodwell crowned the victory by powering into the area and sending a precise finish beyond Van der Sar to end United's hopes of a comeback.
But it was Everton who took the lead when Fellaini broke up play, fed Drenthe and once Baines delivered into the box it was set for Gibson to smash in, with the shot taking a slight deflection off Gareth Barry.
Tuncay and Afonso Alves were causing plenty of problems on the counter at the other end, but it was Everton creating the better chances and Boro were fortunate when Yakubu went down under Turnbull's challenge, only for referee Howard Webb to wave away the penalty appeals.
The appeals were strangely muted from Everton, but TV replays suggested the Chelsea defender was lucky to get the decision.
The hosts went ahead with a thumping drive from Dmitri Likhtarovich but Marouane Fellaini drew Everton level from a Leighton Baines free-kick and Tim Cahill won it with 13 minutes remaining.
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