Rangers put their relegation fears aside to claim success in the third-round replay at The Hawthorns when Bothroyd netted in the 76th minute.
Redknapp was pleased with the striker, who recently returned from a loan spell at Sheffield Wednesday, as well as keeper Green, who pulled off a string of fine saves.
"Jay Bothroyd did well tonight," he told QPR's official website.
"He was at Sheffield Wednesday but he didn't do very well and they sent him back. I put him out there and asked him to show what he can do."
He added: "Rob Green made a fantastic save in the first half and then a number of others when he came under pressure after we scored.
"That's three clean sheets in a row, which is encouraging. We're picking up some good results now and some clean sheets.
"We're getting the best out of the team at the minute. There are areas that I'd like to improve on, but I can't fault the effort of the players."
Redknapp says QPR's main focus is the Premier League, where they are rock bottom of the table, but admits it is encouraging to be winning games.
"The league is the most important thing. That is the key for us. You've got to try and stay up. The rewards for doing so are so big," he said.
"But winning is a good habit. We've won twice away from home now after not doing so for over a year."