Bothroyd, back from a loan spell at Sheffield Wednesday, settled the tie when he headed home a corner from Alejandro Faurlin.
But he was denied the chance of a reunion with the Owls as they were beaten by MK Dons, who will now visit Loftus Road in the fourth round.
It was another encouraging result for Harry Redknapp's side after collecting four points from their previous two Barclays Premier League with Chelsea and Tottenham.
They also celebrated their first win against Premier League opposition in the competition for 18 years.
Bothroyd was a constant threat but Rangers were indebted to several fine saves from goalkeeper Robert Green for securing their place in the last 32, including a late stop to deny Graham Dorrans.
West Brom's slump in fortunes continued and they have now won only two of their last 11 matches in all competitions and their season is in danger of fizzling out.
James Morrison tried in vain to unlock the QPR defence while centre-back Gareth McAuley had to make several crucial challenges to halt QPR on the counter-attack.