The Toffees got their noses in front on the stroke of half-time when Peter Crouch knocked in an unlucky own-goal for the Potters but fortunately, Jerome netted within two minutes of coming on for Tony Pulis.
David Moyes' men, fresh off a 4-0 home drubbing of Fulham on the weekend, had to deal with the ever present aerial threat provided by Stoke but for the most part were able to defend with discipline until the 69th minute when Jerome encroached Tim Howard's goal.
Everton's opener came about when Tony Hibbert's cross late in the first half glanced off Crouch and onto teammate Marc Wilson before deflecting back off the striker and in past Asmir Begovic in quite calamitous circumstances.
With just over 20 minutes left, Jerome got on the end of a Matthew Upson pass from Stoke's defensive half to beat off John Heitinga and Phil Jagielka with a powerful run into the area before seeing his shot bobble off Jagielka and past Howard for the eventual leveller.
Nikica Jelavic and Steven Pienaar were extremely close to grabbing an opener with their attempts after the half-hour mark which saw the Croatian's deflected effort go just inches over the crossbar from Magaye Gueye's cross before the South African midfielder's vicious dipping shot had Begovic beaten but was slightly off target.
Stoke had some solid chances of their own, especially in injury time when Kenwyne Jones volleyed over from close range following a goalmouth scramble before Howard managed to keep out Robert Huth's deflected free-kick to allow his side to grab a point.
Britannia Stadium, Stoke-on-Trent - 26,500