Andrew Surman and Junior Stanislas both got on the scoresheet to help Eddie Howe's side record their first Premier League away victory of the season.
Stoke boss Mark Hughes insisted during the week his side were not in dire straits this season but they were quickly in this one as the away side raced into a two-goal lead.
Surman opened the scoring in the 16th minute before Stanislas doubled their advantage just two minutes later from the penalty spot after a rusty-looking Ryan Shawcross brought down Benik Afobe.
Stoke came out a different side in the second half and finally breached the Bournemouth defence in the 63rd minute with Mame Biram Diouf turning in after a long ball forward fell to him in the box.