Loic Remy opened the scoring after 14 minutes but Julio Cesar's fumble allowed Gaston Ramirez to level on the stroke of half-time.
QPR rallied after the break and Bothroyd prodded home late on to earn an invaluable three points.
"They answered all the questions," Redknapp said of his players. "They worked their socks off today.
"We worked hard last week against Man United. It was just they had a little bit too much for us.
"We couldn't have worked much harder. We had 14 shots at goal last week and we gave them a game, but they were better than us.
"Today we have come out and worked just as hard again and no-one could have worked harder than the players I took away on the trip."