Comic and magician John van der Put, who performs as Piff the Magic Dragon, ventriloquist Paul Zerdin and Stevie Starr, The Professional Regurgitator, made it through.
Van der Put and Zerdin hail from London, and Starr from Glasgow.
They join a line-up of 10 competing for the top prize of $1m (Â£645,000) and their own Las Vegas show.
Van der Put, who often performs with Mr Piffles, a Chihuahua in a dragon costume, secured his place by performing card tricks with the panel of judges.
He was the opening act for Mumford and Sons on their UK tour, recently performed his show Breakfast at Piffany's at the Edinburgh Festival and has already had a Las Vegas residency.
Writing on Twitter following his win, he said: "Thank you America for #votingPiff.  I believe I am feeling something you humans call happy. See you in six days."
During the semi-final, Zerdin used a mask to turn judge Howie Mandel into a human ventriloquist's dummy, getting him on stage to "audition" for the other judges and making him "twerk".
He is already an established performer in the UK who has appeared twice at the Royal Variety Performance.
Starr, The Professional Regurgitator, regurgitated smoke bubbles and a raw egg in his semi final performance.
He is no stranger to TV talent shows having taken part in Britain's Got Talent in 2010 as well as the equivalent shows in France, Italy, Germany and the Czech Republic.
Comedian Drew Lynch, music duo The CraigLewis Band and singer Benton Blunt are among the other finalists.
The show's finale, featuring former Spice Girl Mel B and German supermodel Heidi Klum as judges, will air in the US on September 15 and 16.
Three British acts have advanced to the finals of America's Got Talent, the US TV talent search.