Esko Kaonpää began his ice hockey career with Tampereen Ilves, winning his first Finnish league title in 1962. After spells with HIFK and Tappara, Kaonpää then joined the Austrian side Klagenfurter AC, winning back-to-back titles in 1970 and 1971. He returned to his native Finland the following year to win a second league title with Tampereen in 1972. During the rest of the decade he played for teams in Austria, West Germany, and Italy. Kaonpää made 21 appearances for the national team including representing them at the 1964 Innsbruck Olympics.