A product of Partizan's youth system, Stevanović made his senior team debut with their affiliated club Teleoptik, before being promoted to Partizan's first team in the summer of 2009.[1] He was loaned to Spartak Subotica in January 2010,[2] before signing a permanent contract with the club in the summer of 2010.
In the 2016 winter transfer window, Stevanović left Novi Pazar and moved to Bosnian Premier League club Željezničar.[3] He won his first trophy with Željezničar in the 2017–18 season, the Bosnian Cup.[4] On 27 July 2019, in an away win against Mladost Doboj Kakanj, he made his 100th appearance for the club.[5][6] Stevanović scored his first goal for Željezničar on 9 November 2020, in a league game against Borac Banja Luka.[7] In June 2021, he decided to terminate his contract with the club, leaving Željezničar after five years.[8]