Needing a victory in the Premier League clash to keep their hopes alive, the Toffees found the net in the 72nd minute through Phil Jagielka. He seemed to have scored the winner, following a corner from Mikel Arteta, only for referee Howard Webb to rule out the defender's effort.
Manager David Moyes was left baffled at the decision, which led him questioning the fourth official as play resumed. It was a pivotal moment not only in the game but also in Everton's season.
It will be of little consolation to the Scot that his side made it 10 Premier League games unbeaten, a club record. Stoke, meanwhile, will take heart from this performance after a 7-0 hammering by Chelsea last weekend.
They managed to keep a clean sheet and could have secured the victory early in the second half.
Rory Delap, however, somehow managed to miss a gilt-edged chance and his effort is bound to be screened over and over again on television.
Everton began brightly and Arteta saw a lot of the ball and delivered an inswinging cross in the 18th minute that was headed clear by Robert Huth.
Back came Stoke and Ryan Shawcross escaped his marker following a long throw from Delap, only to head wide.
It was an open game and Victor Anichebe tested Asmir Begovic, on his full debut, with a shot from distance before Ricardo Fuller failed to get enough power on his flick and Tim Howard made the save.
Stoke supporters screamed for a penalty in the 30th minute when Glenn Whelan went to ground following an aerial collision with Arteta but Webb waved play on.
Everton responded and Phil Neville released Victor Anichebe three minutes later, only for the striker to drag his effort wide of the post.
There was little to choose between the sides at this stage and Whelan found space on the edge of the area but fired his effort high over the bar.
Everton then surged forward and Leighton Baines saw his powerful shot beaten away by Begovic after he linked up with Arteta.
Stoke should have taken the lead in the 48th minute but Delap was guilty of an astonishing miss. Mamady Sidibe started the move by picking out Matthew Etherington on the left and he quickly played the ball towards the back post.
Delap had timed his run perfectly but with only Howard to beat somehow managed to steer his effort the wrong side of the post.
Stoke were anxious to make the breakthrough and Whelan tried his luck from distance in the 55th minute but his effort failed to test Howard in the Everton goal.
There was a sense of anticipation three minutes later around the ground when Whelan shaped to take a free-kick.
It was awarded after Baines had bundled over Fuller but Whelan only succeeded in hitting the wall with his effort.
This was a good spell by Stoke and Fuller dragged a shot wide after being set up by the industrious Sidibe.
Everton hit back and Jagielka found himself in a decent position after 67 minutes. However he was unable to get enough power on the shot.
Then came that controversial moment for Everton and Stoke responded when Whelan set up Fuller 20 yards out but he sent his shot high over the bar.
Everton kept pressing forward but failed to get the victory needed to secure a Europa League place.