István Szívós Sr. started swimming in 1931 at the Szeged Swimming Association and when the family moved to the capital in 1932 he was also a footballer for a short time in the BESZKÁRT youth team. That same year he became an athlete at the Hungarian Swimming Association (MÚE). He started playing water polo in 1935, and two years later he was a member of the team that won the second division, earning him an invitation to the national team. In 1940 he transferred to Magyar AC. In 1942 he was on the front line for a year and then in 1943, he won the Hungarian water polo championship. In 1945 he was the founder of the Budapesti Vasas water polo team and played for Vasas between 1945 and 1946, and 1949 and 1958. In the interim period he was a player for Vörös Lobogó and MTK. He won the Hungarian championship once with Vasas, in 1953.
Szívós participated in three Olympic Games during his career. He won a silver medal with the national team at the 1948 London Olympics, and won Olympic gold medals at the 1952 Helsinki Olympics and the legendary 1956 Melbourne Olympics. He also competed in swimming in Melbourne and was a member of the winning water polo team at the 1954 European Championships in Torino. He was also a two-time Hungarian champion in river swimming. He played for the water polo national team sixty-four times.
In 1957 Szívós began coaching at his former club Vasas and then became the coach of MAFC in 1959. In 1962 he obtained a coaching and then a professional coaching diploma. Between 1964 and 1966 he was the national captain of the Egyptian water polo team and then in 1967 he became the head coach of the newly formed OSC (Medical University SC), with which he won the Hungarian championship twice and the Hungarian Cup once. From 1970 he worked as a professional consultant for the club. In addition to his coaching work he also worked as a referee.