Yakubu scored his first goal of the season as Everton stretched their unbeaten run in the league to five.
The Nigerian striker showed neat control before lashing into the roof of the net after Tim Cahill's driven shot had rebounded off an upright.
Stoke defended stoutly in the main, and Tuncay thought he had grabbed the opener - but was adjudged to have fouled Leighton Baines in the process.
However, Everton generally produced more quality than their opponents.
A contest that was played at a lively pace throughout weathered some dull passages of play either side of half-time, but came alive at a Stoke free-kick just past the hour mark when the ball rebounded off a melee of players and trickled towards Baines and Tuncay.
Disallowed goal should have stood - Pulis
The Turkish midfielder, playing in a supportive role just behind striker Kenwyne Jones, was quicker to react than the Everton left-back, but referee Lee Probert adjudged he had unfairly impeded Baines before bundling the ball into the net.
Within a minute, Abdoulaye Faye got some part of his body in the way of a fierce Tim Cahill volley as one of a series of Everton attacks again failed to come to fruition.
Immediately after that, Everton boss David Moyes sent on a second striker in Louis Saha to assist Yakubu with Seamus Coleman - who had had a disappointing afternoon on the right wing - withdrawn.
And around five minutes later, the hosts finally got the goal their more purposeful football had threatened.
Following Yakubu's strike, gleefully celebrated by Moyes who has stood by the 27-year-old despite his scoring drought, Stoke - who had never gone into an overly defensive shell - reacted with some bright football of their own.
Jermaine Pennant and Glenn Whelan came on - but Matthew Etherington who was putting in a good performance on the left side of midfield remained their most productive player.
A decent effort from him late on flashed past Tim Howard's far post.
Stoke's greatest fault throughout was their inability to force saves from the big American goalkeeper - best summed up by two incidents that could have seen them opening the scoring early in the second half.
First, Marc Wilson ignored pleas from Etherington and Tuncay for a pass and instead cut inside his marker to blast a wayward shot harmlessly wide.
Then centre-half Ryan Shawcross also failed to find the target when hitting a right-foot volley goalwards from just inside the area following Roberth Huth's cross.
Moyes praises Stoke after 'welcome' victory
That period of pressure from Stoke suggested they might be the team to break the deadlock - and when a series of decisions from Mason went against the home team the crowd started showing its frustration.
In that sense, Yakubu's goal could not have come at a better time - but to suggest their win was undeserved would be to stretch a point.
Everton had been much the better team in a first half devoid of interest. Yakubu, the otherwise excellent Steven Pienaar and Neville all failed to make the most of decent opportunities in the first 45 minutes.
Stoke were unable to make that profligacy count in their favour, and Tony Pulis's side are now winless in their last three games.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.