Though only 17 years and 8 days of age, Džajić was handed his first team debut by head coach Miša Pavić in a Yugoslav First League match against FK Budućnost Titograd on 8 June 1963, which finished 0–0. It was the last week of the 1962–63 league season with FK Partizan already clinching the league championship and Red Star hovering in 7th place, out of European spots. The match was played at Omladinski Stadium at Karaburma because Red Star's famous home Marakana was in the final construction stages.