Djurovski started out at Red Star Belgrade, making his senior debut in 1979, aged 16. He spent a total of seven seasons in their first team, winning two national championships (1981 and 1984) and two national cups (1982 and 1985). In the summer of 1986, Djurovski made a surprising and controversial move to Red Star's bitter rivals Partizan. He stayed for four years at Stadion JNA, including an inactive season because of his compulsory military service. Some of his most memorable performances with the "Crno-beli" include a 1989–90 European Cup Winners' Cup tie against Groningen, which eventually secured him a transfer to the Dutch club in 1990. In his later years, Djurovski went on to play for several Slovenian clubs.