János Dunai first played for Bajai Bácska in the third division and then continued his career in the first division Pécsi Dózsa. He scored 70 goals in 266 league matches for Pécs.
Dunai played for the Hungarian national team once in 1960 in a friendly match against England. That year he was also a member of the Olympic team, with which he eventually won the bronze medal at the Roma Games. He played a major role in the national team’s success as he scored twice against Peru and France in the group matches, while the Hungarian team won the match against Italy for third place with his goal.
In 1964 Dunai earned a law degree from the Janus Pannonius University of Pécs, and in 1974 he earned a football coaching diploma from the College of Physical Education. As a coach he managed Pécsi MSC for several terms, but also worked at PVSK, Sellye, Keszthely, and Szigetvár. His brother Antal Dunai was also a national football player.