Cahill applied a trademark headed finish to Mikel Arteta's 77th minute corner after City had battled back from two goals down.
Everton boss David Moyes fielded £15million debutant Marouane Fellaini, while Cahill made an earlier than expected return from injury.
But it was Stoke, who included newcomer Ibrahima Sonko at the heart of defence, who made most of the early running.
A Liam Lawrence free-kick was blocked by the defensive wall and Ricardo Fuller's shot was well saved by Tim Howard after Lawrence created the chance.
Everton saw more of the ball as the pace of Stoke's early efforts abated, but they were struggling to make any inroads against a well drilled defence.
That all changed four minutes before the interval when a touch of class from Yakubu put them in front.
Arteta and Fellaini were involved before Yakubu moved the ball on to his left foot and struck a low shot beyond Thomas Sorensen from 14 yards.
It looked all over six minutes after the interval when Victor Anichebe was allowed the freedom of the Britannia Stadium to glance in a header from Arteta's free-kick.
But Stoke launched a storming recovery which brought them back on level terms and could have put them in front.
Seyi Olofinjana crashed in a 16-yard volley after Howard could only punch Rory Delap's long throw into his path in the 55th minute.
And another spearing long throw from Delap led to Phil Jagielka heading the ball into his own net just eight minutes later.
Fuller walked the ball into the net in between times, but the goal was controversially ruled out for a foul on Joseph Yobo.
But Everton rode the storm and, after Moyes was sent to the stand for his protests over what should have been a penalty, Cahill notched the decider.
Stoke gave everything they had in search of an equaliser with Fuller, substitute Mamady Sidibe and Danny Higginbotham all going close.
But Everton held on for their second win of the season and left Stoke in the bottom three.