Dražić started playing football with FK Obrenovac 1905, and later was with Red Star Belgrade youth selections for two years.
He joined Javor Ivanjica in summer 2013, and scored his first goal for new club on his SuperLiga debut, against Red Star Belgrade.
[12] During the autumn same year, Dražić also scored a goal against Mladost Lučani on 20 August and against Inđija in a cup match played on 21 September 2016.
After Ilija Stolica replaced Bratislav Živković as a coach of FK Voždovac, Dražić started next two matches from the bench.
[13] Returning in the first squad, Dražić scored a goal and noted an assist for a 2–1 victory in away match against Borac Čačak, played on 30 November 2016, being also nominated for player of the fixture.
[14] Dražić was elected as a man of the match in 4–1 victory against Vojvodina on 1 April 2017, scoring a goal and being fouled for a direct free kick.
[19] Dražić joined China League One club Changchun Yatai in July 2019, providing an assist on his debut, a 3–1 away victory over Nei Mongol Zhongyou.