Yakubu (left) and Cahill each scored in Everton's win
Everton fended off a determined Stoke fightback to earn a win courtesy of Tim Cahill's header.
Yakubu had clinically sidefooted the visitors ahead before Victor Anichebe's glanced header added a second.
But two Rory Delap long throws were not dealt with by Everton and Stoke drew level through a Seyi Olofinjana volley and Phil Jagielka's headed own goal.
Everton boss David Moyes was sent to the stands after his side were denied a penalty before Cahill headed in.
Everton will feel justice was done, while Stoke will be left ruing the lack of reward for their valiant efforts.
They had shown some steely character and aerial threat to haul themselves back into the game as they continued to serve notice that the Britannia Stadium will prove a testing venue for most of the top-flight.
Stoke had gone behind after 41 minutes of tame action in which only Ricardo Fuller's saved shot was the main threat of a goal.
Stoke's disciplined ranks had comfortably held Everton at bay as the visitors looked short of the invention needed to engineer a breakthrough.
But the stalemate was broken when Stoke's backline failed to clear a long ball and allowed Yakubu to score.
The Nigerian took two touches as he shifted the ball on to his left foot before nonchalantly striking home a shot.
Everton were rewarded for their patience and would have added to their lead on the counter-attack only for backtracking striker Dave Kitson to block a Cahill shot just before half-time.
No regrets for Moyes after sending off
Another lapse at the back then cost Stoke when Olofinjana let Anichebe glance in a Mikel Arteta free-kick from the right.
But Olofinjana atoned for his lapse in concentration and breathed new life into the match when he volleyed in after Delap's long throw was not cleared by the Everton defence.
Everton knew they would have to cope with the aerial threat of the home side as well as Delap's long throws but dealing with them proved another matter and they suffered a torrid time.
Stoke upped their intensity and Fuller thought he had equalised when he rounded Howard and slotted in.
But his effort was ruled out as referee Alan Wiley had already blown his whistle for Fuller fouling Joseph Yobo in winning possession.
Pulis positive after running Toffees close
Stoke's anger at the harsh decision was calmed when another Delap long throw was flicked on by Jagielka into his own net past the despairing Tim Howard.
Everton were then on the receiving end of what appeared a wrong decision when Leon Cort seemed to handle the ball in the penalty area only for the visitors to earn a free-kick just outside the box.
Moyes vented his fury at the assistant referee and was pointed to the stands by Wiley.
He then watched from seats behind the dug-out as Cahill headed in Arteta's corner to earn his side a win which moves them into the top half of the table.
Stoke boss Tony Pulis: "We started well in the first 20 or 25 minutes we were fine, but we faded away and gave them the impetus.
"The first goal was disappointing but it was a good finish.
"We had a bit of a pop at half time and after that Everton scored and it looks like game over. But fair play, the players kept battling and kept going."
Everton boss David Moyes on referee Alan Wiley's non-award of the penalty after Leon Cort's handball: "The referee pointed to the spot and where I was standing he (Cort) was two yards inside (the box).
"If I was wrong I would apologise to him and if he was wrong I would expect the same."
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