Kevin Mirallas`s first-half solo effort was all that separated the two sides during an even contest at Goodison Park.
Victory for Everton keeps their hopes of European qualification alive, but defeat for Stoke leaves them looking over their shoulder at the relegation zone.
It`s now one win in 25 games away from home for the Potters and they lie just four points above the drop zone after Wigan`s win earlier in the day.
The Potters made a bright start and after winning a throw-in deep inside the Everton half they should have taken the lead.
Ryan Shotton`s trademark long missile was flicked goalwards by the returning Robert Huth but after Tim Howard had palmed the ball back into the danger zone, Jon Walters proceeded to somehow turn his effort onto the crossbar when a goal looked inevitable.
Stoke continued their bright start, but were maybe fortunate not to concede a penalty after 12 minutes. A wayward Everton shot stuck Marc Wilson on the arm but it was adjudged to have been unintentional by the referee and his assistant.
With Everton playing three at the back, Wilson was being afforded time and space to forage forward down the left and was unfortunate not to see one his deep crosses converted by either Cameron Jerome, Peter Crouch or Walters who were packing the penalty area.
Everton had their first real opportunity midway through the first-half when Geoff Cameron`s tame header allowed Leighton Baines to whip the ball back into the middle, but Nikola Jelavic headed straight at the grateful Asmir Begovic.
Despite being the brighter of the two sides, the Potters were deal a sucker punch on the counter attack after 28 minutes. Tim Howard punched clear a free-kick and with Steven Nzonzi unable to stop Mirallas` surging run into the Stoke half, he shimmied past Cameron and struck his effort under the onrushing Begovic.
Mirallas almost made it two seconds later. Steven Nzonzi lost the ball deep inside his own half, but the Belgian international could only screw his shot narrowly wide of the far post.
Stoke almost drew level, though, when another long throw from Ryan Shotton caused mayhem inside the Everton penalty area. The ball fell to Glenn Whelan on the edge of the area and his deflected effort struck Crouch who, with only the keeper to beat, was wrongly adjudged to be offside.
Walters was the next City player to try his luck from just inside the box, but his shot was comfortably palmed away to safety by the agile Howard who was returning to the Everton team after a recent spell on the sidelines.
Everton started to control the second half which resulted in chants of ‘Attack! Attack! Attack, Attack, Attack!` from the travelling supporters and their wishes of a goal almost came on 55 minutes, but Huth directed his header just wide from a deep Whelan free-kick.
Ryan Shotton missed an even better chance five minutes later. After marking a darting run to the far post, he diverted his header over the crossbar with the goal gaping.
Stoke were starting to dominate proceedings but were nearly dealt another damaging blow, however Begovic got fingertips to Jelvaic`s curling free-kick as the game edged towards the final 15 minutes.
Tony Pulis made a double substitution by brining on Kenwyne Jones and Charlie Adam to try and give his team fresh impetus in search of a leveller but it failed to materialise as the Potters lost again on their travels.