A Europa League chasing City side arrived on Merseyside to face a buoyant Everton, still buzzing after cementing their FA Cup final place and holding Chelsea to a draw in midweek.
Everton could have easily recorded a big victory with a little more accuracy from City loan ranger Jo, who was missing from the line-up against his employers.
Segundo Castillo kept his starting place after his impressive game against Chelsea, while Moyes rotated his defence with Phil Jagielka replacing Joleon Lescott.
Pablo Zabeleta missed out through injury but otherwise Mark Hughes' side was unchanged from the team that beat West Brom convincingly last time out.
The game started tentatively with both midfields cancelling each other out. Robinho had a penalty shout waved away early on for City, while at the other end Steven Pienaar almost put Louis Saha through.
Robinho forced a smart tip over from Tim Howard on 25 minutes as he began linking well with Elano.
Castillo and Stephen Ireland were having a right old battle in midfield with one momentous tussle resulting in the ball falling to Pienaar, who delivered a pin point cross low into the box which Saha just failed to get a touch on to.
Leighton Baines effected a fantastic sliding tackle on Robinho to prevent a certain goal with the fleet-footed Brazilian through on Howard's goal.
Everton's reprieve was short-lived though as Elano played a lovely straight ball from midfield to Robinho, who twisted and turned to escape the attentions of Jagielka.
When this ploy failed the striker simply faced the defender head on and shot home past Howard with almost desultory ease to send the away fans into ecstasy.
Saha should have levelled minutes when Tony Hibbert delivered a perfect cross which the Frenchman contrived to head well over.
After the Toffees fantastic but emotionally draining week, Moyes would find it difficult to lift his team at half-time, while Hughes asked his boys for more of the same.
Everton forced a corner in the opening minute of the second half. Castillo's glanced header from Pienaar's kick rolled inches wide to the chagrin of the Goodison faithful.
Minutes later Marouane Fellaini almost scored a highly impressive equaliser as he received a low Pienaar cross, which he flicked up and caught on the turn, however Shay Given produced a great reaction save to palm clear.
Ireland added a second goal against the run of play as Felipe Caicedo played a short ball to Robinho, who floated a high ball over the Blues' defence to an unmarked Ireland, who made no mistake from eight yards out as he guided the ball past an onrushing Howard.
Moyes had seen enough by the 60th minute and hauled off Saha, Fellaini and Castillo for Tim Cahill, Dan Gosling and James Vaughan.
By 65 minutes though disaster struck as the inspirational Jagielka stretched for a ball as what looked like his right ankle gave way under him.
The defender was stretchered off and with all Everton's subs used it was down to a ten-man home side to attempt to claw back some kind of result.
The game was becoming increasingly bad tempered and after an altercation between Cahill and Vincent Kompany, Elano was lucky to only see yellow after an horrendous tackle from behind on Pienaar.
Deep into stoppage time Gosling pulled a consolation back for Everton as they piled forward, but City held firm to record only their second away victory of the season.