© PA Photos
Rangers quickly went down the other end and Bothroyd forced Boaz Myhill into action with a stinging drive.
After half time Lukaku remained the hosts' biggest threat as he again tested Green with a near-post shot on the hour.
Harry Redknapp's side took the lead with 15 minutes on the clock when Alejandro Faurlin lofted the ball over for Bothroyd to nod home.
The Baggies pressed for an equaliser in the late stages but couldn't find the net as Green twice denied Markus Rosenberg.
QPR will take on MK Dons in the fourth round after Karl Robinson's side saw off Sheffield Wednesday this evening.