After starting out at his hometown club Priboj, Dragaš moved across the border and played for Bosnian team Lokomotiva Brčko in the 2009–10 season.
He subsequently returned to Serbia, spending two and a half years with Metaloplastika, before moving to Macedonian club Vardar in November 2012.
[1] In the 2013–14 season, Dragaš rejoined Metaloplastika, helping them reach the EHF Challenge Cup final.
In June 2014, Dragaš signed a three-year deal with Handball-Bundesliga team Bergischer HC.
[3] After playing for Metalurg Skopje and Hapoel Rishon LeZion, Dragaš moved to Romania and joined Székelyudvarhelyi KC in June 2017.