The former Barcelona forward gave Mark Hughes' side the lead 38 minutes into a stop-start first half with a calmly taken penalty, though Everton missed a host of gilt-edged opportunities to level before the break.
Stoke, who were lucky not to have boyhood Evertonian Jonathan Walters sent off in the first half, were content to play on the counter in the second period, with Roberto Martinez's side ultimately proving toothless in attack.
Stoke shot-stopper Asmir Begovic did well to save the efforts that did come his way during a subdued second half, with Leighton Baines the only Everton player able to make any headway on the Bosnian's goal.
Both teams struggled to find their rhythm from there, with Ryan Shawcross and John Stones receiving lengthy spells of treatment for bangs to the head and ankle respectively, although Stoke could have taken the lead had Mame Biram Diouf's effort not been blocked by Phil Jagielka.
Everton burst into life at the other end just after the half-hour mark, when Mirallas' corner landed at the feet of Gareth Barry at the back post, only for the midfielder to somehow shank his effort wide when it seemed easier to score.
The controversy from that decision continued when, from the resulting free-kick, Stoke countered through Bojan, who burst into the box before being pulled back by James McCarthy – Mason this time pointing to the penalty spot.
Bojan, who seemed to have gone down softly, took the penalty himself and slotted the ball calmly into the bottom-left corner, only just evading Tim Howard's despairing grasp.
Thanks to the earlier injuries, there was 10 minutes of first-half stoppage time, during which Steven Naismith almost brought Everton level with a header that Steven N'Zonzi hacked off the line.
Samuel Eto'o was brought on soon afterwards as Martinez attempted to add craft and impetus to Everton's forward line, but Stoke's high pressing game continued to thwart them as the visitors saw out the win.
