The west Londoners out-battled Southampton on their own patch on Saturday, with Loic Remy putting them ahead after just 14 minutes.
Gaston Ramirez equalised after a mistake by goalkeeper Julio Cesar, but QPR secured just their third Premier League win of the season thanks to Bothroyd's strike.
Next up is a home clash with Sunderland and the striker, who spent the first half of the season on loan at Sheffield Wednesday, knows his side need to kick on.
"The gaffer has been great," Bothroyd said of Redknapp. "He fills the boys with confidence even when you're not playing.
"There is that buzz around the training ground and he always talks to you.
"I've come in as Bobby Zamora was injured. Obviously I've scored but it was one of those where it was all about the squad and you do as well as you can and I was delighted to get a goal.
"Every game is a cup final. We can't complain and we have to be up for every game.
"We won and we have a massive game next week. if we take our foot off the pace, then this game means nothing."
As well as an important team performance, Bothroyd was proud of his own effort.
The 30-year-old has featured just twice for the Hoops in the league this season and was glad to show Redknapp what he can offer.
"It's great to score goals and it was always about getting the win," Bothroyd said.
"The gaffer has done a lot with us this week on tactical things and we played a counter-attacking game and played well.
"We had the better chances I thought and we're happy and delighted.
"It worked well for both goals and it was a terrific team performance. Everybody worked really hard.
"The back line was really tight and obviously we took it to them and we have a smile on our face, which hasn't happened too much this year."
The victory moves QPR within four points of safety and brings Southampton back into the relegation dogfight.
QPR boss Redknapp believes Saints are safe but that means little to defender Jose Fonte, who was disappointed by the way Saturday's match panned out.
"We dominated the game," the centre-back said. "We felt we had the game under control for the full 90 minutes.
"The fact is they had two chances and scored two goals.
"There was not enough movement and not enough passes up front.
"They came, they sat back and defended well. It was up to us create chances and we didn't create enough.
"It is very disappointing but we need to move on because we have another big game we need to take points from.
"We will learn lessons from that and try to go to Norwich and get points.
"It is hard to take this but we need to move on. There is no other way to take it."
While the 2-1 loss was disappointing, Saints fans will have been relieved to see Fonte return to first-team action after a knee complaint.
"Personally, I felt good," the defender added. "The knee didn't hurt me.
"I felt strong so at least it is a positive that I didn't feel any pain in my knee."
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