Kašlík was born in Poličná in Moravia, Austria-Hungary (now part of Valašské Meziříčí in the Czech Republic).
In 1940 he began working for Brno radio as a conductor, and made his debut as a director at the Prague National Theatre with Boleslav Vomáčka's The Watersprite.
In 1945 he and Alois Hába founded on the grounds of the former Neues Deutsches Theater the Grand Opera of the Fifth of May (today the State Opera Prague).
In 1953 he returned to the Prague National Theatre, establishing international reputation as a director.
His use of television, film, and projections brought him wide acclaim, particularly due to the collaboration with Josef Svoboda.