The 14-year-old, from the village of Hesleden, County Durham, received a standing ovation for her energetic rendition of Tina Turner’s River Deep - Mountain High during the first part of the two-night final.
Following her performance on Tuesday, judge Howie Mandel told Courtney she is a “superstar in the making”.
He said: “Amazing. Your letters were on fire. You are on fire. You are amazing, you are a superstar. I love everything about you, I love your look, I love your style, I love your voice.
“I love your performance. I think the world does too, we are watching a superstar in the making right here on America’s Got Talent.”
In the earlier rounds, Courtney had performed covers of classic songs including Steppenwolf’s 1969 hit Born To Be Wild, Otis Redding’s Hard To Handle and James Brown’s Papa’s Got A Brand New Bag.
Mandel’s fellow judge, Simon Cowell, said her choices of song made her a “trendsetter”.
He said: “I always say the most important part of being a star is being remembered and that’s what you do. Every time you perform, people are going to remember you.
“You are going to inspire people. You are going to tell a whole new generation of people what rock and roll is all about. All these great songs.
“So you are more than a great singer, you are like a trendsetter. Without people like you, we don’t look good.”
Mel B told Courtney she had “killed it” and described her as a “free bird”.
She added: “You were wild and crazy and at one time I thought your legs were going to snap but it was brilliant, original, you’re amazing.”
The British teenager previously took part in The Voice Kids UK, and is able to compete in America’s Got Talent as the competition is open to anyone with a visitor visa to the US.
Courtney is one of 10 acts in the America’s Got Talent season 13 final and the second part is set to begin at 1am in the UK on September 20.