Antal Újvári started his sports career as a football goalkeeper in the colors of ВТК, then moved to Elektromos TE where he kept goal for ten years. He won several amateur championship titles with his team. During his football career, he also played handball in the colors of Elektromos for three years.
In the 1936–1937 handball season he won the Hungarian championship title with his team. He was also selected to the national team in this sport.
In 1936 he appeared in the Hungarian national team 5 times. In 1936, as a member of the national team, Újváry participated at the Olympics in Berlin. He also served as a handball referee after the Olympics.
He later played as a football goalkeeper in lower division teams (Fodrászok TK, Díjbeszedő SK).