Arnošt Klimčík played his entire career from 1964-77 with one club – Baník Karviná, winning Czechoslovakian titles in 1968 and 1972. Internationally, Klimčík was on the Czechoslovakian national team at the 1967 World Championships, where they won the gold medal and at the 1972 Olympics, where they won a silver medal.
After his retirement from sports, Klimčík, a graduate of Secondary School of Mining in Ostrava, worked as a deputy head of the hydraulic department of First May Coal Mine (now Barbora Coal Mine) in Karviná. He later worked as a project supervisor with Road and Railway Constructions Ostrava, later named EUROVIA CS.