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.
Everton were unbeaten in five Premier League home games prior to this rain-swept clash, but despite a fast-paced start they were sluggish in possession.
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.
Everton started brightly in a frenetic opening period and they could have been awarded a penalty inside the first minute when Kevin Mirallas was bundled over by Bojan, but referee Lee Mason waved away the appeals.
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 home fans were appealing for a red card 10 minutes prior to the interval when Walters tugged down Baines as the England full-back bore down on goal, but Mason was lenient and chose to only book the Republic of Ireland striker.
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.
The drama did not stop there either, as, with just a minute of the additional period remaining, Mirallas fired into the side-netting despite having plenty of time and space to hit the target from 10 yards.
Martinez had never lost to Stoke in his managerial career, but maintaining that record was made more difficult when he was forced into two changes at the start of the second period, with injured duo Jagielka and Howard having to be replaced by Antolin Alcaraz and Joel Robles.
Despite plenty of possession within the opening 15 minutes of the second half, Everton were unable to trouble Begovic, with Mirallas, Naismith and Ross Barkley largely anonymous.
They finally did manage to carve out an opportunity through Baines in the 64th minute, as the full-back cut inside before drilling in a low right-footed effort that Begovic did well to parry away down low to his right.
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.
