Had the Blues made it six in a row it would have been the first time such a sequence had been recorded since 1987 - but it wasn't to be - especially after Robert Huth had given the Potters an unlikely second half lead.
Captain Leon Osman promptly equalised but his side couldn't muster a winner in a game which they dominated.
David Moyes was without Joseph Yobo and Steven Pienaar but his side didn't lack defensive steel or creative influence in the opening period.
The returning James Beattie's third minute header was one of few first half chances for the Potters, but Everton soon threatened Stoke's goal with a long range Jack Rodwell effort.
Tim Cahill was next to threaten as he sought to add to his midweek goal in Belarus. The Australian fired wide after collecting Marouane Felliani's pass and then moments later nodded straight at Thomas Sorensen.
After Ricardo Fuller had a rare headed chance for the visitors, Everton took a stranglehold on the game, with Rodwell shooting over and Diniyar Bilyaletdinov claiming a penalty when Ryan Shawcross appeared to handle his venomous shot in the box.
Even Tony Hibbert got in on the action with an effort that was blocked away, then in the 38th minute Osman went closest yet.
The stand-in skipper turned smartly on the edge of the box and produced a stinging drive that Sorensen managed to palm over.
Cahill again went close from the corner and John Heitinga curled a free-kick over at the end of an opening period that the Toffees dominated.
So it was totally against the run of play when Huth headed in Matthew Etherington's corner in the 50th minute to give Stoke the lead.
Osman's first left-footer was Everton's immediate attempt at an equaliser and it went harmlessly wide.
His second, shortly after, was perfectly executed and curled delightfully into the top left-hand corner of the goal to level the scores in the 55th minute.
Bilyaletdinov sent in a couple of speculative efforts and Cahill again headed at Sorensen as Everton pushed for a winner.
And with 15 minutes remaining, Moyes bolstered his frontline, introducing Jo and Yakubu in a bold switch for Osman and Bilyaletdinov.
But it was Fellaini who went close next, again from range, his shot was deflected off target.
The remainder of the game proved a frustrating period for the Blues as they struggled to fashion clear chances against the physical and disciplined Stoke backline.
So when the ball bell to Yakubu in injury time Goodison held its breath - but the striker's shot was weaker than he would have liked and Sorensen easily saved.
