The away side started well but will rue their early missed chances, as Bournemouth grew into the game and eventually took the lead.
Following an even start, QPR began to take control of the game. First, after 15 minutes, Jermaine Jenas saw his long-range effort tipped wide by the goalkeeper, before Benoit Assou-Ekotto got forward but fired the ball straight down the middle.
Bournemouth improved after that and started to create chances, although Yann Kermorgant will be disappointed with himself for wasting several of them.
The hosts eventually took one of their chances, as Elphick rose highest to head home Ian Harte's corner in first-half stoppage time.