Luigi De Canio's side thoroughly deserved their victory and would have won by a bigger margin had they taken all their chances.
Rangers made the perfect start, going ahead after just four minutes.
Vine crossed from the right and Dexter Blackstock's lay-off found Agyemang, who lifted the ball over keeper Heinz Muller before applying the finishing touch with his head.
Barnsley responded to that early setback well and should have equalised when Michael Coulson headed Marciano van Homoet's cross wide of the target.
But on the stroke of half-time, the Tykes found themselves further behind.
Vine combined brilliantly with Agyemang, twice exchanging quick passes with the striker before tapping in his first goal since his loan move from Birmingham was made permanent.
Martin Rowlands almost netted an even better goal just after the break.
The Rangers captain weaved his way past three defenders and rounded Muller only for his shot to be cleared off the line by Dennis Souza.
Defender Matt Connolly - one of four QPR players making their home debuts - sent a long-range chip just wide and Mikele Leigertwood's drive was saved by Muller as the home side continued to dominate.
Barnsley did have their chances, however.
Fitz Hall's error put Jamal Campbell-Ryce through on goal but he shot straight at Rangers keeper Lee Camp.
And substitute Kayode Odejayi missed the visitors' best opportunity when he blazed wide from Campbell-Ryce's right-wing cross.
The R's win was slightly marred by what looked like a leg injury to defender Damion Stewart, who was stretchered off late on.