During his career, he won 32 doubles titles and finished runner-up an additional 29 times, including at the French Open in.

Slozil and his partner, Renata Tomanová (also from Czechoslovakia), won the 1978 French Open mixed-doubles championship, defeating Virginia Ruzici (Romania) and Patrice Dominguez (France).

The mixed doubles championship was an important event in those days, contested by top players, with John McEnroe and Mary Carillo having won the year before.

He was a member of the winning Czech Davis Cup team in 1980, along with teammates Ivan Lendl, Tomáš Šmíd and Jan Kodeš.

Shortly after the start of his tenure as Graf's coach, she had her Grand Slam breakthrough year in 1987, winning six tournaments and then defeating Martina Navratilova in the French Open.