Nash, 86, and his 82-year-old wife Alicia were killed when their taxi crashed in New Jersey, they said.
The mathematician is renowned for his work in game theory, winning the Nobel Prize for Economics in 1994.
His breakthroughs in maths - and his struggles with schizophrenia - were the focus of the 2001 film.
Russell Crowe, who played him in the film, tweeted: "Stunned... My heart goes out to John & Alicia & family. An amazing partnership. Beautiful minds, beautiful hearts."
Alicia Nash helped care for her husband, and the two later became prominent mental health advocates.
'Beautiful Mind' mathematician John Nash killed in crash
