Born in 1931 in Jajce (Bosnia), Dusko Goykovich studied at the Music Academy in Belgrade from 1948 to 1953. As a youth he played with several jazz and dixie bands, mostly for dancing audiences and eventually on parties at the embassies of the capital. When the 18-year-old joined the Radio Big Band of Belgrade, he was considered a talented young jazz man who also can read music. When he left the band five years later, he had grown into a fine big band player and featured soloist. Dusko went to Germany where he quickly became an integral part of its uprising young jazz scene. In 1956 he made his first record as a member of the Frankfurt All-Stars.
After a short stint in the big band of Munich's Max Greger, Dusko stayed for four and a half years with Kurt Edelhagen's band, then Europe's leading jazz orchestra. Francy Boland, Claus Ogermann, Jerry van Rooyen and Rob Pronk were among the arrangers who worked for Edelhagen. In addition to being the band's premier trumpet soloist, Dusko performed with such as Stan Getz and Chet Baker. It came as no surprise when in 1958 he was invited to play with the Newport International Youth Band at the Newport Jazz Festival. Other members of the Newport band included Albert Mangelsdorff, Ronnie Ross, George Gruntz, and Gábor Szábo.
Following the performance at Newport, Dusko's trumpet became very popular in Europe. In 1961 the Berklee School of Music offered the 29-year-old a grant for studying composition and arrangement in Boston where Herb Pomeroy was to become one of his teachers. Looking forward to writing his own arrangements for his great love, the big band, Dusko concentrated on his studies at Berklee so exclusively that he regrettingly turned down offers by Count Basie, Stan Kenton and Benny Goodman to join their bands. While at Berklee Dusko (now also on flugelhorn) recorded with the Berklee School Quintet and Orchestra including fellow students such as Gary Burton, Mike Gibbs, Sadao Watanabe, Steve Marcus, Mike Nock, and Dave Young.