During his playing career that spanned 21 seasons from 1960 to 1981, Ćurković made over 700 official appearances for three clubs in Yugoslavia and France, winning numerous trophies and reaching the European Cup final on two occasions, with Partizan in 1966 and Saint-Étienne in 1976.

During the 1960–61 Yugoslav First League, his first senior season, Ćurković played 13 (out of 22) games and helped the team narrowly avoid relegation from the top flight.

He would become the first-choice goalkeeper in his debut season after Milutin Šoškić joined the army to complete his compulsory military service, helping the club win the championship.

[5] Between 1989 and 2006, Ćurković served as president of his former club Partizan, establishing a long-lasting partnership with general secretary Žarko Zečević and sporting director Nenad Bjeković.

[12] He was also a recipient of France's Legion of Honour during a ceremony held in Belgrade on 5 October 2005, in which his former Saint-Étienne teammate Michel Platini presented him the award.