Draper, the British No 1 and fifth seed, lost in four sets: 5-7, 6-3, 6-2, 6-4.
Bublik's victory ends Britain's hopes in the singles at Roland Garros, following Cameron Norrie's earlier loss to Novak Djokovic.
Alexander Bublik, 27, has become the first Kazakh man to reach a Grand Slam quarter-final.
Bublik acknowledged the significance of his win, calling it the best moment of his life and expressing his determination to prepare for the next match.