The 23-year-old forward, a high-profile close-season arrival from Barcelona, wasted little time in making a positive impression in his first appearance at the Britannia Stadium on Wednesday.
After left-back Erik Pieters had put Stoke ahead against opposition relegated from Spain's top flight last term, Bojan ended the first half with a flourish, finding the net for the third time in as many appearances courtesy of a thunderous strike from the edge of the area.
Betis hit the woodwork twice in the second half, but were unable to find a goal as the hosts recorded their first victory in pre-season at the sixth attempt.
In addition to Bojan, fellow new signings Phil Bardsley and Mame Biram Diouf also started, with Mark Hughes choosing to field a strong XI that could yet take the field in Stoke's Premier League opener against Aston Villa next Saturday.
Betis forward Nono tested Asmir Begovic inside the first minute, but Stoke soon settled and saw both Bojan and Diouf threaten prior to Pieters opening the scoring on the half-hour.
Following neat build-up play involving Bojan, the ball was worked to Bardsley on the right and his low cross was turned home by fellow full-back Pieters at the far post.
Vincenzo Rennella came close to an equaliser within four minutes, dragging a shot narrowly wide, before Bojan took centre stage on the stroke of the interval.
The former Barca man picked up a loose ball and beat a Betis player before letting fly with a venomous strike into the top corner of the net.
Ruben Castro should have halved the visitors' deficit nine minutes after the break, but fired wildly over having been left free inside the box.
Bojan and Peter Odemwingie, who was introduced at the start of the second half together with Steve Sidwell and Geoff Cameron, then came close to extending Stoke's advantage before Betis struck the post twice in quick succession through Jorge Molina and Cedric Mabwati.
Molina saw another opportunity go begging amid a predictable raft of substitutions, while Bojan was afforded a standing ovation when he came off in the closing stages.
