Ireland midfielder Gibson left United to move to Goodison Park earlier this month and his fierce strike on the hour, which took a slight deflection, proved decisive.
United, who trail their rivals on goal difference, eased past Stoke thanks to two penalties at Old Trafford from Javier Hernandez and Dimitar Berbatov.
City and United have 54 points from 23 matches, five ahead of Tottenham Hotspur who thumped Wigan Athletic 3-1 at White Hart Lane.
Dejected City manager Roberto Mancini said he had underestimated mid-table Everton.
"Probably, probably, probably," the Italian said in an interview with Sky Sports after he was asked if he had taken Everton too lightly.
"Maybe it's my fault because I prepared, maybe, bad for this game... it is very important that we win our next game against Fulham."
The leaders enjoyed plenty of possession but created few openings to trouble a resolute home defence.
Everton, who paraded new signing Nikica Jelavic at half-time after the Croatia striker's move from Rangers, almost broke through early on but Joleon Lescott was well positioned on the goal-line to clear Denis Stracqualursi's close-range header.
City went closest to an equaliser when midfielder Samir Nasri rattled the crossbar with a swerving 30-metre piledriver on 36 minutes.
The match was halted for around four minutes late in the first half when a fan went on to the pitch and handcuffed himself to a goal-post.
Police freed the man who had attached himself to the post in the goal that City were defending.
NO MISTAKE
United had no problems overcoming Stoke despite handing a first league start to goalkeeper Ben Amos, called up due to David de Gea being ill and Anders Lindegaard injured.
With Wayne Rooney still out with an ankle injury, Sir Alex Ferguson partnered Berbatov alongside Javier Hernandez up front and both made no mistake from the spot.
Mexico striker Hernandez fired home on 38 minutes after Jermaine Pennant had felled Park Ji-sung.
United were awarded a second penalty eight minutes after half-time and this time Berbatov scored after Antonio Valencia was brought down.
Spurs cemented third place after Gareth Bale scored twice against bottom side Wigan.
Bale's eighth and ninth league goals of the campaign came either side of a Luca Modric goal. Substitute James McArthur got a late consolation for Wigan.
A stoppage-time own goal by Neil Taylor rescued a point for fourth-placed Chelsea, on 42 points, who escaped from Swansea City's Liberty Stadium with a 1-1 draw.
Things looked bad for Andre Villas-Boas's side who trailed to a first-half goal from their former player Scott Sinclair and had defender Ashley Cole sent off with four minutes remaining.
But Portugal defender Jose Bosingwa then got lucky in the fourth minute of added-time when his shot took a big deflection off Taylor to beat goalkeeper Michel Vorm.
Sinclair's goal was the first Chelsea have conceded in five games in all competitions.
Liverpool striker Andy Carroll ended his goal drought to help fire the Anfield side up one place to fifth, above inactive Arsenal, after a 3-0 victory at struggling Wolverhampton Wanderers who remain in the bottom three.
Carroll, who has failed to impress since his big-money move a year ago, broke the deadlock seven minutes into the second half with his first goal in 12 league games before Craig Bellamy and Dirk Kuyt rewarded Liverpool's dominance.
