Former Manchester United midfielder Gibson lashed home his first goal since quitting Old Trafford earlier this month as the Premier League leaders slipped up on Merseyside once again.
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.
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 ruled any 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 Sylvain Distin and Phil Jagielka, coming in at centre-half in place of 20-year-old Shane Duffy.
On-loan Argentinian Stracqualursi was given another chance up front as Louis Saha, with rumours linking him with a deadline day move to Tottenham, was missing from the teamsheet.
City welcomed back influential captain Vincent Kompany after a four-match suspension but the frustration they have felt over two cup exits and numerous decisions in recent weeks only continued.
Stracqualursi, who scored his first Everton goal in last Friday's FA Cup win over Fulham, went close to doubling his club tally when he forced Joleon Lescott to head off the line with five minutes gone. He pleaded for a goal after beating Joe Hart with a powerful header from a Leighton Baines corner but replays showed former Everton man Lescott had cleared.
Sergio Aguero brilliantly controlled a long David Silva pass to create an opportunity on the edge of the Everton box but shot wide, before Everton were awarded a free-kick 30 yards out after Gareth Barry clattered into Tim Cahill - but Kompany headed away Landon Donovan's ball into the box.
Hart was incensed after being bundled over by Stracqualursi but Walton had awarded him a free-kick before Donovan blazed over.
After Kompany was booked for pushing over Cahill and Marouane Fellaini had twice gone close, City almost opened the scoring as Nasri struck the bar with a crisp 30-yard shot. Aguero raced in for the rebound but was flagged offside.
Play was held up just before the interval when a man ran into Hart's goalmouth at the Park End and handcuffed himself to one of the posts. He was removed by police and led away after a short struggle but Walton played five minutes of stoppage time.
City started the second half well with Nasri whipping in a low cross which just evaded Edin Dzeko before the Bosnian then headed straight at Tim Howard.
Aguero then had a shot deflected behind but Lescott was penalised for a foul on Howard at the resulting corner.
Everton made City pay for their failure to take advantage of their possession as they made the breakthrough just before the hour.
Royston Drenthe forced the visitors back before Baines found Donovan in space with a deep cross into the box.
The American calmly laid off the ball for the inrushing Gibson, who fired in from the edge of the box - via a deflection off Barry.
City tried to respond as Nasri got to the byline but his ball in bounced onto the bar and behind, before Gibson volleyed narrowly wide from 20 yards moments later.
City appealed for a penalty for handball after substitute Aleksandar Kolarov hit the ball at Neville but Walton saw nothing untoward.
City were almost caught on the break as they protested but Micah Richards got back to deny Donovan.
Mancini's men poured forward as time ran out but were unable to find a way through, with Nasri having their final effort - a shot which was deflected into Howard's arms.