The new arrival from Barcelona, who signed a four-year deal at the Britannia Stadium on Tuesday, was named in Mark Hughes' starting line-up on Saturday but exited at the interval with his side 1-0 down courtesy of Daniel Adlung's strike.
Bobby Wood doubled the Bundesliga 2 outfit's lead against a much-changed Stoke side after the break, with goalkeeper Gabor Kiraly keeping a hard-earned clean sheet.
Bojan had an opportunity to make a swift impact on his debut, but his shot from inside the box flew wide in the fifth minute, while Marko Arnautovic's free-kick drew a good save from Kiraly.
Stoke enjoyed plenty of possession but lacked a cutting edge in the final third, something they were made to rue when Adlung fired a 26th-minute effort past Thomas Sorensen from 25 yards.
Wood was finding plenty of space down the left but the American wasted several opportunities to provide a telling delivery, affording Stoke the chance to get to the break level.
Hughes rung the changes at the interval, with Bojan among the many to make way, but he did not get the response he was looking for.
Though Break Shea provided two presentable openings for Charlie Adam and Peter Crouch, both were missed and the German side capitalised in the 51st minute as Wood tapped in from close range.
Late pressure from Hughes' men saw Kiraly deny Steve Sidwell, Jonathan Walters and Adam as Munich recorded a deserved victory.
Stoke, who drew 1-1 with Borussia Monchengladbach on Sunday, face one further German test, against Schalke on Tuesday, before returning to England next week, while 1860 begin their league campaign against Kaiserslautern on August 4.