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."