The Toffees had gone in front in the 44th minute at the Britannia Stadium thanks to an own goal from the hosts' former Liverpool striker Peter Crouch, whose unfortunate last touch sent the ball over the line after Tony Hibbert has delivered a cross into the box.
Jerome levelled in the 69th minute, making a forward burst and holding off two defenders before firing past Tim Howard.
While seventh-placed Everton will be disappointed not to make off with all three points, they have extended their unbeaten run in the league to seven games and, perhaps even more satisfyingly, their lead in the table over Merseyside rivals Liverpool - who are eighth and lost 1-0 to Fulham on Tuesday night - to three points with two games remaining.
Stoke, meanwhile, are up a place to 13th, two points away from the top half in which they are aiming to finish for the first time since their promotion to the division in 2008.
Rory Delap - back in the Potters side fresh from signing a new contract extension - delivered an early trademark long throw which Robert Huth met, only for the referees' assistant to raise their flag, and at the other end Sylvain Distin nodded a corner wide.
Stoke's Jon Walters was next to try to connect with a delivery, this time from Matthew Etherington, but his reward was a collision with Distin which led to him receiving treatment and having his head heavily bandaged.
Etherington then lofted an effort over before Huth headed another Delap missile too high.
Everton were struggling to handle the hosts' aerial bombardment, with Howard just able to fist a cross away from Kenwyne Jones' head, but when the Toffees pushed forward themselves, Leon Osman managed to get a couple of shots away in quick succession, both of which were saved by Asmir Begovic.
After Delap caused another scramble with a throw which Huth met to prompt Howard to punch the ball behind, the visitors went on the attack again, a cross from Magaye Gueye - who had come on a substitute in the 22nd minute for the injured Victor Anichebe - cannoning off Nikica Jelavic and over.
Steven Pienaar then saw his fierce strike helped over the bar by Begovic, Gueye drilled wide from the resulting corner, and moments later, Everton snatched the lead.
It came in freak circumstances a minute before the break, Pienaar playing a short free-kick to Hibbert, whose cross struck Stoke's Marc Wilson and then his team-mate Crouch before bouncing past Begovic.
A quiet start to the second half was punctuated by a succession of penalty appeals which amounted to nothing, while Osman fluffed his lines after being teed up Pienaar.
Stoke boss Tony Pulis refreshed things by deploying his three substitutes in one fell swoop in the 67th minute as Jerome, Ricardo Fuller and Glenn Whelan replaced Walters, Crouch and Delap, and his decision paid off almost immediately as the Potters pulled level.
Jerome showed great determination to power forward and he got the better of Phil Jagielka and John Heitinga to force the ball past Howard and in.
Jelavic attempted to restore the visitors' lead in the final 10 minutes but miscued and Stoke might have won it themselves late on, Jones shooting over and Howard getting down to save a deflected Huth free-kick.