David Moyes' team of happy travellers recorded a seventh away league win of the season to regain fourth place from their Merseyside rivals.
But City's second home league defeat of the campaign was a big blow to their own hopes of bringing European football to the club next season.
After their derby win at Manchester United in their last league game a fortnight earlier it was back to earth with an almighty bump for Sven-Goran Eriksson's City side, who suffered back to back home league defeats after losing their unbeaten home record to Arsenal.
Eriksson not surprisingly went with the same starting line-up which won at Old Trafford for the first time in 34 years, though this display was a pale shadow of that performance.
It was Everton who took a first-half stranglehold on the game establishing a two-goal lead which could conceivably have been four.
Everton had a penalty shout rejected for a handball appeal against Micah Richards, while Steven Pienaar hit an upright and that was before they took the lead on the half hour.
Yakubu played the ball out wide left to Tim Cahill and then ran onto the return to lift a shot into the roof of the net from six yards for his 16th goal of the season.
Everton doubled their lead eight minutes later when Lee Carsley retrieved an over-hit corner and swung over a far-post cross which defender Joleon Lescott headed home for his eighth goal of the campaign.
Eriksson introduced Elano at half-time and then in the second half brought on strikers Felipe Caicedo, for his home debut, and Nery Castillo.
But it was to no avail as Everton held on to their lead and were rarely troubled and had three great second-half chances to increase their margin of victory.
The hard working Cahill saw a close-range effort cleared off the line by City captain Richard Dunne.
Everton were then denied a penalty after a handball by Nedum Onuoha, with referee Rob Styles adjudging that the offence had taken place just outside the box.
City keeper Joe Hart then made a brilliant save to keep out Carsley's free-kick.
City had to wait until the 85th minute before forcing Everton keeper Tim Howard to make his first save of the match, a brilliant stop to deny Castillo.
City's miserable night was completed in stoppage time when Martin Petrov was sent off for stupidly aiming a kick at Leon Osman after the ball had gone.