The Potters came from behind to earn a valuable point after Ryan Shawcross' own goal was cancelled out but there was no punishment for the Toffees talisman's headbutt
An own goal from Ryan Shawcross broke the deadlock just before half time, the defender flicking a Steven Pienaar cross beyond Asmir Begovic in the Stoke goal to give the visitors the lead.
The Potters were level seven minutes into the second half after Kenwyne Jones rose to meet a Shawcross ball and head past Tim Howard, while Marouane Fellaini was lucky to stay on the pitch late in the game after headbutting the Stoke captain.
Stoke welcomed back Charlie Adam after his one-game suspension, with the Scot replacing Dean Whitehead in the centre of midfield, while Andy Wilkinson filled in at the back for the suspended Ryan Shotton.
A hamstring injury to Kevin Mirallas meant that Everton boss David Moyes had to make one change to the side that beat Tottenham, with Steven Naismith the man chosen to fill the Belgian's boots.
In a competitive first half both sides had half-chances to open the scoring but they lacked the accuracy to test either goalkeeper. Darron Gibson and Seamus Coleman carved out an opportunity for Nikica Jelavic in the 10th minute but his low shot from inside the penalty area was well blocked by Shawcross.
The deadlock was broken in the 36th minute with a wonderful header from Shawcross; unfortunately for the Stoke skipper, it was into his own goal. Neat build-up play from the visitors down the left created a chance for Pienaar to cross and as Shawcross attempted to clear, his flicked header flew past Begovic and into the top corner.
The hosts were level seven minutes into the second half when Jones punished bad play from both Phil Jagielka and Howard. It appeared the goalkeeper would easily save the Trinidadian's header from Shawcross' excellent ball but he got his bearings all wrong and allowed the effort to squeeze inside the post.
The game then had its most controversial moment as a Marouane Fellaini headbutt on Shawcross went unpunished by referee Mark Halsey. The two men were jostling for position inside the Stoke box, waiting for an Everton corner, when the attacker turned and lashed out at the Stoke captain, but he escaped any punishment.
The introduction of Cameron Jerome and Peter Crouch gave Stoke more impetus in the closing moments of the game but Howard did well to save from both men from dangerous situations to preserve a point.