The presidential palace said Herzog died Friday, 62 years after he and Louis Lachenal made it to the peak of Annapurna in the Himalayas without the aid of oxygen.
President Francois Hollande said in a written statement the trek was "an exploit that will remain in our collective memory."
CNN said Herzog did far more than rest on the laurels of his 1950 exploit. He was minister for youth and sport under President Charles de Gaulle and was a successful businessman.
Herzog spent his later years in the Paris suburbs but will be buried in the city of Chamonix, where he served as mayor from 1966 to 1977, Radio France Internationale said.