Kozubayev previously played for FK Šiauliai and FK Granitas in Lithuanian A Lyga. On the last day of the 2015–16, winter transfers window, Kozubayev signed with the Serbian club FK Jagodina.[2] After his Jagodina contract expired, Kozubayev signed a contract with Kyrgyz side Dordoi Bishkek on 17 August 2016.[3]
On 28 January 2019, Dordoi Bishkek confirmed that Kozubayev had left the club after his contract had expired.[4]