Born in Sarajevo, PR Bosnia-Herzegovina, FPR Yugoslavia to a Croat father Franjo Bregović and Serb mother Borka Perišić, Goran grew up with an younger sister Dajana and younger brother Predrag. Their father was from Croatian Zagorje, specifically Sveti Petar Čvrstec village near Križevci, while their mother was born in Virovitica to parents that had shortly before her birth arrived in the nearby village of Čemernica, settling there from the village of Kazanci near Gacko in eastern Herzegovina. Her father, Goran's maternal grandfather, fought in the Royal Serbian Army at the Salonica Front during World War I and as a reward received land in Slavonia where he soon moved his family.
Goran's parents met shortly after World War II in Virovitica where his mother Borka lived and his father Franjo (who fought on the Partisan side during the war) attended a Yugoslav People's Army (JNA) military school. Franjo Bregović soon got his first job, teaching ballistics at a military school in Sarajevo, so the couple that at the time moved there. Goran, their first child, was born in 1950 in Sarajevo.
Goran was 10 years old when his parents divorced. In later interviews, he mentioned his father's alcoholism as the reason for the breakdown of their marriage. Soon after the split, his father moved to Livno, taking Goran's younger brother Predrag with him while Goran remained living with his mother in Sarajevo, visiting his father and brother every summer in Livno. Their father soon retired and eventually moved back to his home village in Zagorje while Goran's brother Predrag later moved back to Sarajevo for university studies.
Goran played violin in a music school. However, deemed untalented, he was thrown out during second grade. His musical education was thus reduced to what his friend taught him until Goran's mother bought him his first guitar in his early teens. Bregović wanted to enroll in a fine arts high school, but his aunt told his mother that it was supposedly full of homosexuals, which precipitated his mother's decision to send him to a technical (traffic) school. As a compromise for not getting his way, she allowed him to grow his hair long.