He became a member of the Orchester Der Wiener Staatsoper in 1936 and then joined the Wiener Philharmoniker in 1939.
Wobisch had been a member of the NSDAP party since 1933 (when it was still illegal in Austria), and became a member of the SS in 1938 after the annexation of Austria into the German Empire (Third Reich). Wobisch was dismissed from Wiener Philharmoniker at the end of World War II, but was allowed to rejoin in 1947. He concealed his past so successfully that he was able to become the orchestra's executive director from 1953 to 1968. Wobisch became a professor at the Vienna Music Academy in 1958. He founded the Carinthian Summer Music Festival (Carinthischer Sommer) in 1969 and remained its artistic director until his death.