The visitors lose their advantage at the top of the table, as the ex-Manchester United player's second-half strike helped his former side pull level on points in the title race
David Moyes' side gave a strong showing, but the first half would be dominated by incidents outside of the football being played on the pitch.
Midway through the opening 45, Nikica Jelavic's transfer to the club from Rangers in a £7 million deal was confirmed, with the Croation being introduced to the crowd at halftime, but not before a fan delayed proceedings by handcuffing himself to a goalpost.
However, Gibson's finely struck effort, which found its way past Hart albeit through a deflection, ensured tomorrow's headlines will remain focused on events on the field.
For the home side, John Heitinga and Tony Hibbert lined up at centre-back, whilst striker Louis Saha did not make the bench ahead of a proposed move to Tottenham. Roberto Mancini welcomed Vincent Kompany back into the City starting XI following his four-game suspension.
The first chance of the half fell to Everton just after five minutes, as Denis Stracqualursi's header following Royston Drenthe's corner beat Joe Hart but was blocked by Joleon Lescott on the line.
It took half an hour before another chance was created, with the home side again getting a sight on goal. A deep ball in from Leighton Baines found Stracqualursi, who headed down for Fellaini, but the Belgian fired the shot wide.
Is it like this everyweek? | New boy Jelavic is introduced to the Goodison Park crowd
Just two minutes later, Manchester City had their first real chance of the match as Samir Nasri fired a shot from 25 yards, beating Tim Howard but hitting the bar, before Aguero's shot from the rebound was well saved, even though the Argentine was offside.
But the talking point of the first half came five minutes from interval when the match was briefly delayed following a disturbance on the pitch, in which a supporter emerged from the Gwladys Street Stand and handcuffed himself to the post next to City keeper Joe Hart.
The fan spent five minutes attached to the post before being removed by police before the game restarted.
The second half resumed at a similar pace until the hour mark, when Drenthe broke into the opposition half before releasing Baines.
The full-back's cross found Donovan, who teed up Gibson after an initial miss-control, with the ex-Manchester United man firing past Hart via a slight deflection to give the home side a 1-0 lead.
Roberto Mancini's side looked to change it around two substitutions, as Adam Johnson and Kolarov entered the game, but it was Everton who continued to press.
Stracqualursi took the ball down the left-hand side and crossed. The City backline easily cleared but only as far as Gibson, who wound up another volley, this time with his left foot, that went just a yard wide of the goalmouth.
There was controversy with 20 minutes remaining, as City had a penalty shout turned down when Kolarov claimed Phil Neville had handled in the box, but the visitors' complaints were waived away.
City continued to push for equaliser but just couldn’t find a way past the solid home side who hung on for a deserved three points at Goodison Park.