The Potters fell behind in the 36th minute through an own goal from captain Ryan Shawcross, who inadvertently nodded in a Steven Pienaar cross.
Jones equalised in the 52nd minute, heading Shawcross' long-range delivery past Tim Howard.
The visitors might have had Marouane Fellaini sent off if the match officials had spotted the midfielder's apparent headbutt on Shawcross in the second half.
The Potters passed up an opportunity just after the half-hour mark. Glenn Whelan swung in a corner and Jon Walters headed the ball back but an unmarked Steven Nzonzi could only nod it into the arms of Howard.
But it was Everton who went ahead nine minutes before half-time, the ball going over Asmir Begovic and in having come off the head of the unfortunate Shawcross, who had been trying to clear Pienaar's cross. Leon Osman should have made it 2-0 but blasted wide from Leighton Baines' cut-back.
Stoke appeared full of purpose after the restart and were level within seven minutes, Jones heading past Howard from Shawcross' long ball into the box.
Having ended his drought, Jones looked in the mood for more. Moments later he dinked the ball over Howard, only to see it hit the woodwork.
Everton then seemed lucky to escape punishment, with Fellaini appearing to headbutt Shawcross as the two players marked each other in the box. The Stoke man was left lying on the turf but the officials took no action.
With 12 minutes remaining, Cameron Jerome found himself through on goal and saw his shot come off the legs of Howard, with fellow substitute Peter Crouch's follow-up lob then being headed away by Sylvain Distin.