The striker's 77th-minute goal earned rock-bottom Rangers a precious three points, but they are still four points adrift of safety and face another crucial game at home to the Black Cats on Saturday.
"Every game is a cup final. We can't complain and we have to be up for every game," Bothroyd said.
"We won and we have a massive game next week. if we take our foot off the pace, then this game means nothing."
Bothroyd thanked manager Harry Redknapp for giving him his first Premier League start of the season, saying: "The gaffer has been great.
"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.
"It's great to score goals and it was always about getting the win."
Bothroyd praised Redknapp's counter-attacking strategy at St Mary's, adding: "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."