Bojan, signed from Barcelona earlier this week, started in a central role behind Mame Biram Diouf and almost opened his account for his new club within six minuetes when he was slipped through by Marko Arnautovic only to fire just wide.
Arnautovic also went close to scoring the opening goal with a low 30-yard free kick that 1860 Munich goalkeeper Gabor Kiraly, formerly of Crystal Palace, tipped around the post.
It was the German side that went ahead in the 25th minute, however, when Daniel Adlung fired home from distance, leaving Thomas Sorensen little chance.
Andy Wilkinson hobbled off before half-time to be replaced by Phil Bardsley, and Mark Hughes made 10 further changes at the interval as Bojan made way.
The alterations made little difference, however, as 1860 Munich added a second six minutes into the second half through Grzegorz Wojtkowiak.
Kiraly was quickly off his line to prevent a shooting opportunity for Steve Sidwell as Stoke looked to reduce the deficit but the Bundesliga 2 side, who begin their league season a week on Monday, comfortably saw out the game as their greater fitness levels told.
Stoke play the third game of their pre-season campaign against Schalke on Tuesday evening.
Stoke XI, first half: Sorensen; Wilkinson, Shawcross, Wilson, Muniesa; Shotton, Whelan, Ireland, Bojan, Arnautovic; Diouf
Stoke XI, second half: Bachmann; Bardsley (Ness), Huth, Teixeira, Pieters; Walters, Nzonzi, Sidwell, Adam, Shea; Crouch