Kojac represented the United States at the 1928 Summer Olympics in Amsterdam. As a member of the winning U.S. team in the 4×200-meter freestyle relay, he received a gold medal. Kojac and teammates Austin Clapp, Walter Laufer and Johnny Weissmuller set a new world record of 9:36.2 in the event final. Individually, he won another gold medal in the men's 100-meter backstroke with a second world record time of 1:08.2. He also finished fourth in the men's 100-meter freestyle in 1:00.8.