Darron Gibson helped former team Manchester United as his first goal for Everton was enough to beat Premier League leaders Manchester City.
Denis Stracqualursi's header was nodded off the line by Joleon Lescott as Samir Nasri hit the bar in the first half.
The hosts paraded at half-time and Gibson lifted the crowd further when he struck following Leighton Baines' cross.
City pressed but though Nasri hit the bar again, they could not capitalise.
With Manchester United winning at home to Stoke, City's failure to get even a point means they are now only top of the Premier League on goal difference.
And the defeat has once again highlighted how Roberto Mancini's team are missing Yaya Toure, despite the return of skipper Vincent Kompany, as they fell to their third loss of the campaign.
Everton's victory was their fourth at home this season, but it continues their good run against City, who have won once in 14 visits to Goodison Park.
A lacklustre first half was given an unusual twist when a supporter appeared to , delaying the game by five minutes.
But as poor as the visitors were for stretches of the game, Everton battled well throughout and Stracqualursi was a tireless presence on his own up front.
The Argentine nearly gave Everton the lead, with Lescott denying him following Royston Drenthe's corner, and he also tested Joe Hart with another header soon after.
But with goals hard to come by at Everton, and striker Louis Saha missing from the squad before , chances were thin on the ground early on.
Their only other chance of note came when Marouane Fellaini fired wide after a well-worked move.
City were gifted their first opportunity when Drenthe carelessly gave away possession after 23 minutes. That enabled David Silva to find Sergio Aguero, but the visitors' top scorer was off-target.
Nasri's 25-yard shot against the woodwork turned out to be their best effort, but the league leaders were largely subdued on a freezing night.
Everton's new Croatian striker Jelavic, signed from Rangers, was paraded on the pitch at the break in an effort to inspire the home crowd, although it was City who began to attack with purpose when play resumed, Aguero having a penalty shout when his driving run was brought to an end.
But it was Everton who took the lead when Fellaini broke up play, fed Drenthe and once Baines delivered into the box it was set for Gibson to smash in, with the shot taking a slight deflection off Gareth Barry.
Goodison rejoiced and Gibson, who had spent most of his career at Manchester United before signing for Everton on 13 January, was understandably elated that his first goal for his new team came against United's arch rivals.
The Northern Irishman almost added a second as he fired wide from 25 yards soon after, but City finally seemed to realise the seriousness of the situation and began to press Everton's goal.
Aleksandar Kolarov was introduced in place of Joleon Lescott and he found room to cross from the left, testing Tim Howard with one effort and also hitting Phil Neville's arm with another to loud penalty appeals.
But for all their pressure, the leaders created few openings, and Nasri's shot which bounced into the ground and onto the top of the bar was as close as they came.
Having scored almost at will during the early part of the season, City's performance here will worry Mancini.
And he will hope an upcoming run of five games against teams in the bottom half of the table can add some zest to a side that, despite its expensively assembled talents, is currently looking short of impetus.