The Toffees may have been feeling the effects of their long haul to Belarus on Thursday, but they were nowhere near their usual combative self.
Stoke took the lead five minutes after the break when Robert Huth headed home his first goal for his new club from a corner - but stand-in skipper Leon Osman lashed home a fine equaliser five minutes later, curling the ball into the top of the net from around 18 yards out.
Everton made three changes to the side that won the Europa League match against BATE Borisov, with Jo, Dan Gosling and Ayegbeni Yakubu reverting to the bench to allow John Heitinga - cup-tied in Europe - top scorer Louis Saha and Jack Rodwell into the starting line-up.
Stoke also made three changes from the side beaten at home to Manchester United last time out, with Glenn Whelan, Liam Lawrence and Dave Kitson replaced by James Beattie, Ricardo Fuller and Salif Diao.
Tony Hibbert reverted to right-back, having done exceptionally well as an emergency centre-back in Belarus, with Heitinga partnering Sylvain Distin.
Stoke were intent on attack, fielding former Everton striker Beattie with Fuller while Matthew Etherington took on Hibbert at every opportunity.
The winger's first cross saw Beattie outjump Leighton Baines before planting a firm header into Tim Howard's arms.
The long ball into the box was the main weapon of both sides, and Everton's response was a couple of efforts blocked from Saha and Tim Cahill, before the Australian should have netted after 15 minutes.
Marouane Fellaini swung in a cross from the left, and Cahill ducked low for a header just six yards out, but Thomas Sorensen managed to clutch the effort at the second attempt.
Howard, faced with a similar downward header a minute later at the other end from Fuller, also struggled to hold the effort cleanly.
Russian winger Diniyar Bilyaletdinov came more into the game on the left as the half wore on, and when he cut inside to fire in a rising shot, the ball crashed into Ryan Shawcross' arm.
Referee Andre Marriner turned down the penalty appeals, the Stoke central defender having not moved his arms from his side.
Fuller was booked for a 35th-minute tackle on Heitinga, and Rory Delap was lucky not to follow his colleague into the book when he tripped Baines in full flow.
Osman then produced a fine snapshot on the edge of the box which Sorensen saw late but did well to turn over the bar.
Diao was next to be booked in an increasingly niggly game, for a foul on Baines. Heitinga lifted the resulting free-kick wide.
Stoke started the second period even more belligerently than the first, hounding Everton in possession.
And the Potteries side were ahead when Etherington's right-wing corner curled into the box and defender Huth soared above Cahill to power home his first goal since his £6million summer move from Middlesbrough.
The lead lasted five minutes. Heitinga came forward from the back and exchanged passes with Osman, who struck a fierce drive from the edge of the box which crashed into the net off the underside of the bar.
Bilyaletdinov saw a shot saved by Sorensen, while Fuller failed to trouble Howard with a low drive at the other end.
With 15 minutes left Everton sent on strikers Jo and Yakubu for midfielders Osman and Bilyaletdinov, with Hibbert taking the captain's armband. Stoke responded by replacing Diao with Whelan.
It was a bold ploy by Everton, but one which could see them caught on the break. But even after that change, they did not make a serious opportunity.
Fellaini and Shawcross were booked with four minutes left for a clash in the centre circle, before Stoke sent on Kitson for Fuller during injury-time.
The visitors were powering forward at the end looking for a winner - but neither side deserved to win a scrappy game.
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