German defender Robert Huth headed the Potters into a surprise lead in the 50th minute but Osman curled in an equaliser from the edge of the area five minutes later to ensure the honours finished even.
The Toffees looked the more likely to go on and win the game but David Moyes' men couldn't carve out a winner, with Tim Cahill wasting the best chance, putting a free header straight at Stoke keeper Thomas Sorensen.
A lack of firepower had prevented Stoke from taking three points on their travels this season and manager Tony Pulis was clearly out to remedy this as he started out with former Everton striker James Beattie partnering Ricardo Fuller in attack.
But the formation had little effect on Stoke's performance as Everton dominated possession for the majority of the game.
Cahill should have opened the scoring within 20 minutes but headed straight at Sorensen from Marouane Fellaini's cross.
Osman then threatened to break the deadlock with a trademark bullet from the edge of the area which Sorensen just managed to tip over the crossbar.
Disaster struck for Everton shortly after the break when Huth out-jumped Cahill and nodded Matthew Etherington's corner home to give Stoke the advantage.
But Osman, who was a constant threat in the Everton midfield, struck back five minutes later, curling home a left-footed strike from the edge of the area after a cute one-two with Dutch defender Johnny Heitinga.
The home side pushed for a winner but it wasn't to be as Moyes' men looked visibly tired after Thursday's Europa League win against Bulgarian outfit BATE Borisov.
Subsitutes Jo and Yakubu both had decent efforts saved by the outstanding Sorensen but it was Cahill who had the best chance when he met Louis Saha's deep cross.
But, for the second time in the game, the Australian could do no better than head straight at the Danish keeper.