Potts, from Port Talbot, south Wales, beat six-year-old singer Connie Talbot, from Sutton Coldfield, who had wowed the judges, who included Simon Cowell.
The winner of the ITV1 programme, presented by Ant and Dec, received a cheque for £100,000 and a spot at the Royal Variety Performance.
Some 11 million people saw the hopefuls perform in the show's live final.
Viewing figures dropped slightly to 10.6 million for the later results broadcast at 2200 BST, overnight audience figures show.
"The whole country has gone Paul Potts potty," said host Ant McPartlin who announced during the show that two million people had cast their vote.
However, ITV has refused to confirm exactly how many phone votes it received.
Potts, 36, is a fan of opera legend Luciano Pavarotti and impressed judges with his version of Nessun Dorma.
He first sang opera at the age of 28 for a karaoke competition when he dressed up as his idol.
Diverse finalists
Speaking to BBC Radio Five Live, Potts said he very nearly did not enter the competition.
"I almost didn't press the submit button when I applied online," he said.
"To get as far as I did - I really can't believe it."
Potts, who was told by Cowell that he would start recording his debut album next week, said: "It really proves that variety isn't dead. There's plenty of variety and lots of talent."
Saturday's semi-final also saw bottle-juggling pair the Bar Wizards go through to the final.
The other finalists were Damon Scott, who performed with a monkey puppet, street dancers The Kombat Breakers, and 11-year-old singer Bessie Cursons.
On Friday, impressionist Richard Bates was removed from the contest af ...