Born in Tetovo, Djurovski joined the youth categories of his hometown club Teteks. He subsequently moved to Red Star Belgrade in 1977. Still a junior, Djurovski made his senior debut for the club on 11 April 1979, coming on as a substitute for Duško Lukić in the first leg of the UEFA Cup semi-final, a 1–0 win over Hertha BSC. He spent the following 1979–80 season on loan at Čukarički, becoming the Serbian League (Group North) top scorer with 15 goals. After returning from a loan spell, Djurovski helped Red Star win the Yugoslav First League in the 1980–81 campaign, scoring five goals in the process. He was an integral part of the team and a fan favorite in the following years, leading them to another league title in 1984.