The Toffees had gone in front in the 44th minute 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, 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 lead in the table over Merseyside rivals Liverpool to three points with two games remaining.
* In Scotland, a goal from Michael Hingdon moved Motherwell within touching distance of a place in the Champions League after a 1-0 Clydesdale Bank Premier League victory over Hearts at Tynecastle.
STOKE: Begovic, Huth, Upson, Shawcross, Wilson, Walters (Fuller 67), Whitehead, Delap (Whelan 67), Etherington, Jones, Crouch (Jerome 67).
EVERTON: Howard, Hibbert, Heitinga, Jagielka, Distin, Anichebe (Gueye 22),Osman (McFadden 89), Fellaini, Pienaar, Cahill, Jelavic, Gueye (Stracqualursi 78).
Referee: Anthony Taylor (Cheshire).