Jaidon Anthony scored one goal and set up the other as Bournemouth inflicted QPR's first defeat of the season to go joint top of the Championship with West Brom.
The 21-year-old winger fired the Cherries into an early lead before providing the assist for Dominic Solanke to make it 2-0 before half-time.
Rangers fought back strongly after the break and reduced the deficit when Sam McCallum netted his first goal for the club.
Bournemouth goalkeeper Mark Travers secured his side victory with a string of vital late saves to deny the visitors a point.
Leaders West Brom's goalless draw with Derby on Tuesday means Bournemouth have the same number of points, goals scored and goals conceded after seven games as the Baggies.