Gibson lashed home his first goal since quitting Old Trafford earlier this month as the Barclays Premier League leaders slipped up at Goodison Park.
The visitors hit the bar in the first half through Samir Nasri but Everton, who earlier saw a Denis Stracqualursi effort cleared off the line, again proved stubborn opposition for City.
Screamer: Darron Gibson lets fly to put Everton ahead
MATCH FACTSEverton: Howard, Neville, Hibbert, Heitinga, Baines, Donovan, Gibson, Fellaini, Drenthe (Baxter 78), Cahill, Stracqualursi (Vellios 88). Subs not used: Mucha, Gueye, Barkley, Forshaw, Duffy.
Man City: Hart, Richards, Kompany, Lescott (Kolarov 68), Clichy, Barry (De Jong 86), Milner (Johnson 62) Silva, Aguero, Nasri, Dzeko. Subs not used: Pantilimon, Zabaleta, Savic, Razak.
Booked: Kompany, Lescott.
Referee: Peter Walton.
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City had won just one of their previous 13 league visits to Everton and the extension of that poor record left them top of the table only on goal difference.
Manager Roberto Mancini was again frustrated by another controversial decision as his team were denied a late penalty for handball by Phil Neville but referee Peter Walton ruledany contact accidental.
It was an eventful contest which also saw a bizarre interruption when a protester chained himself to a goalpost.
Everton were buoyed by their capture of striker Nikica Jelavic from Rangers, which was confirmed during the game and the Croatian was paraded at half-time.
David Moyes reshuffled his back line with veteran full-back Tony Hibbert, in the continued absence of SylvainDistin and Phil Jagielka, coming in at centre-half in place of 20-year-old Shane Duffy.