Born to a family of railway workers,[3] he is best remembered for being the first Hungarian and first player from Eastern Europe to win a Grand Slam singles title, at the 1947 French Open (where as the fifth seed he beat Yvon Petra, Tom Brown and Eric Sturgess).

[4] He remains the only Hungarian male player to win a Grand Slam singles title.

[5] Asbóth also reached the semifinals at Wimbledon in 1948 (beating Sturgess and Brown, then losing to John Bromwich).

[6] Hungary's Communist government had let him leave the country only after the personal warrant of the Swedish King Gustaf V that Asbóth would return to his homeland and wasn't going to emigrate.

Asboth won the Open de Nice Côte d'Azur or Nice French Riviera Open tournament in 1947 defeating Bob Falkenburg and again in 1948 defeating Budge Patty and Jaroslav Drobný in the semifinal and final.