Gojković was born on 14 October 1931,[1] in Jajce, Bosnia and Herzegovina, at the time part of Kingdom of Yugoslavia.

[1] In 1961, Gojkovic received a scholarship to attend Berklee College of Music, where he studied with Herb Pomeroy.

In the following years, he performed with Miles Davis, Dizzy Gillespie, Gerry Mulligan, Sonny Rollins, Duke Jordan, and Slide Hampton.

Two years later, Gojković celebrated his 75th birthday with a grand concert in Belgrade, the capital of the Republic of Serbia.

The same year, and recorded and titled to Duško Gojković & Big Band RTS featuring Martin Gjakonovski, Latin Haze, was released by PGP RTS in Serbia in 2014, while a year later it was issued in Germany on Enja with different cover and one track missing.