[1] Born in Zaječar,[2] Kostić passed throw through the Radnički Niš youth categories[3] and later started his senior career with Timočanin in the 2013–14 season,[4] making 7 appearances in the Serbian League East.
[3] Kostić made his official debut for new club in the 17 fixture match of the season against Dunav Prahovo,[7] when he was substituted in after Ivan Bulajić got a red card.
[8] In the next fixture match, against Sloga Despotovac, Kostić made his first league start for Radnički Pirot.
[15] After Ivan Bulajić earned an injury at the beginning of 2016–17 Serbian First League, Kostić became starting goalkeeper from the 3rd fixture match against Mačva Šabac.
[18] Kostić joined Volos on 23 July 2023, He made his first league appearance on 24 November, against OFI Crete which ended in a 1–1 draw.