The Magic Circle, seen as the most famous magic society in the world, confirmed Berglas died in London on Friday night.
Also known as the International Man of Mystery, he was the first magician to have his own programme on British TV, Meet David Berglas, in 1954.
His son Marvin said he was a "giant in the magic fraternity".
In the 1980s Berglas had a second television series - called The Mind of David Berglas - where he entertained celebrities, including Omar Sharif, Christopher Lee and Britt Ekland.
Throughout the decades, he appeared frequently on British TV and radio and became a household name for his stunts, one of which included driving a car around London while blindfolded.
Berglas was renowned for a trick called the Berglas Effect, with which he could find a spectator's chosen card at any number in a deck of cards.
It is regarded as the holy grail of magic effects, the secret of which he took to the grave.
The magician was born to Jewish-German parents who fled the country when the Nazis took power in the 1930s.
After the war, when he was involved with US military intelligence, he ended up in Bradford where he studied textiles, according to theJewish Bradford heritage group.
Mr Berglas was appointed MBE in 2018 for his services to magic and psychology.
He said at the time: "I am delighted to accept this honour but even more pleased that the art of magic has at last been recognised.
"I have spent over 60 years entertaining people in person, on radio and television - 'reading their minds' - but I certainly didn't see this one coming."
His son Marvin said: "My father was a giant in the magic fraternity, known for his originality, creativity and showmanship.
"His mysteries have not only baffled audiences but also his peers. However, to us, his family, he will always be best remembered as a loving husband, father and grandfather."
Illusionist Derren Brown hailed Berglas as "one of our greatest living magical performers" when Berglas was awarded his MBE.
"Generations of magicians owe him a debt of gratitude," Brown said.
"Each of my shows is indebted to his artistry and astonishing body of work. I thank him for his constant inspiration."
David Berglas, one of the most influential magicians and mentalists of the 20th Century, has died aged 97.