Born in Frystak, he studied at the conservatory in Brno (JAMU) and the Academy of Music in Prague (AMU). Trained in bassoon, he played in two orchestras simultaneously: Czech Chamber Orchestra and National Theatre of Prague, and later in the Czech Philharmonic and the Filmorchestra-Concert (FOK). As a student of the famous conductor Rafael Kubelik he soon started his career as conductor at Karlin Theatre Prague where he remained for 20 years. Brazda achieved great success as guest-conductor for Every Man Opera in New York with Porgy and Bess.
He became prominent in the 1950s and 1960s in Prague as an accomplished arranger for Supraphon. His lush string arrangements for popular artists including British singer Gery Scott, German pianist Igo Fischer and Czech singer Karel Gott are highly regarded. His orchestra later became known as Dalibor Brazda Magic Strings.
Brazda also conducted the German cast recording of Fiddler on the Roof in the Mid 1960s (released as Anatevka on CBS, Teldec and Decca) with the Grosses Musical-Orchester und Chor des Operettenhauses Hamburg, for which he won a Gold Record Award. Instead of returning to Prague at the end of the Anatevka-Performances in Hamburg, he emigrated to Dietikon (Switzerland, near Zurich), where he stayed till his death in 2005. In 1985 he became Swiss.
In 2001 he was honoured with the first Cultural Award of Dietikon for his work in musical education of young people and for his musical direction for 25 years of a youth wind-orchestra (Stadtjugendmusik Dietikon) and the Stadtmusik Dietikon, a wind orchestra also, for 15 years.