Unlucky Bournemouth were beaten by an opportunist goal by striker Paul Furlong.
The Cherries looked the better side for much of the game and hit the woodwork twice, but Furlong's header was enough to give Ian Holloway's men the three points.
Bournemouth dominated the first half and Stephen Purches was unfortunate not to score when he first over after pouncing on a mistake by Gino Padula.
The impressive Purches then beat the offside trap and pulled the ball back from the right for Derek Holmes, whose right-footed effort hit the crossbar.
Purches continued to cause Rangers problems and in the 26th minute he crossed from the right for Holmes to head over at the far post.
But for all Bournemouth's dominance, they had a lucky escape in first half injury time.
Kevin McLeod crossed from the left and Karl Broadhurst headed into his own net only for an offside flag to deny Rangers an interval lead.
The R's were much improved in the second period and goalkeeper Neil Moss saved well at his near post after McLeod jinked his way past two men but was unable to find a way past the Cherries stopper.
Broadhurst went close to scoring an own goal for a second time when he sent the ball just over his own bar when trying to intercept Martin Rowlands' cross.
Then on 63 minutes Rangers made the breakthrough. McLeod left Jason Tindall for dead and centred from the left for Furlong, whose close-range header gave Neil Moss little chance.
QPR substitute Kevin Gallen saw a shot saved by Moss and another effort fly wide as Rangers went in search of a second.
But the home team got out of jail again four minutes from the end when Marcus Browning sent a scorching 25-yard effort past goalkeeper Chris Day and on to the bar.
It summed up a frustrating afternoon for Bournemouth, although one bright spot for them was the return to action of Alan Connell.
Connell, sidelined for ten months with a knee injury, came on as a late substitute for Holmes.