The defeat leaves the champions 15 points behind arch-rivals Manchester United in the race for the Premier League title.
Everton took the lead on 32 minutes, midfielder Leon Osman firing past Joe Hart superbly from just outside the area.
Everton were reduced to ten men on 61 minutes, when Steven Pienaar was dismissed for a second bookable offence, after a reckless tackle on Javi Garcia.
Everton’s defence put in a solid display and in injury-time the Toffee’s doubled their lead. Marouane Fellaini strode forward and found substitute Nikica Jelavic who took on Gael Clichy before hit a shot that took a deflection off the Frenchman and beat Joe Hart in the City goal.
The victory moved Everton temporarily up to fifth place in the Premier League and City’s title hopes look all-but finished.
Manchester United stretched their lead at the top of the table with a 1-0 victory over struggling Reading at Old Trafford. The only goal of the game came in the 21st minute, as a Wayne Rooney effort took a hefty deflection off Reading centre-back Alex Pearce and beat Stuart Taylor.
The defeat means that Reading are now seven points away from safety, after rivals Aston Villa defeated rock-bottom QPR 3-2 at Villa Park. The Hoops took the lead through a Jermaine Jenas effort on 23 minutes.
The home side equalised in first half injury-time through Gabriel Agbonlahor. Villa took the lead just before the hour mark through a strike from Austrian star Andreas Weimann. QPR wouldn’t lie down though and on 73 minutes Andros Townsend equalised with a deflected effort.
There was still time for Villa to score a winner with nine minutes remaining. Weimann found striker Christian Benteke and Belgium international fired home. The victory took Villa six points clear of the relegation zone.
Other Premier League results:
Southampton 3-1 Liverpool
Stoke 0-0 West Brom
Swansea 0-2 Arsenal
Are Aston Villa now safe from relegation?
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It's been great seeing Messi and Neymar play together. Hopefully they'
Genius starts with "G" and finishes with "S"...................... So
Lets hope these two greats of the game get a chance to play to gather.
hi. a lot of football expertss expected west bromwich albion to go dow
RVP are you being serious?? A blip in leauge form and a poor finish fo