Born in Sombor, Dražić spent his formative years with Teleoptik Zemun and Partizan Belgrade.
[1] He subsequently went on trial at Russian club Rubin Kazan, before returning to Serbia and signing his first professional contract with OFK Beograd at the age of 18.
[2] Dražić made his official debut for the club on 21 September 2013, coming on as a second-half substitute for Aleksandar Čavrić in a 0–2 away loss to his former side Partizan.
He scored his first professional goal on 19 October, netting his team's first in a 2–3 home loss against Radnički Kragujevac.
On 7 August 2015, Dražić signed a five-year deal with Spanish La Liga side Celta de Vigo.