Cheberko is a product of the UFK Dnipro and FC Dnipro youth systems.[1][2]
On 24 July 2016, Cheberko made his debut for FC Dnipro, coming on as a second half substitute in a match against Volyn Lutsk. Several months later, on 30 April 2017, he achieved another milestone by scoring his first professional goal against the same team. It came via a penalty kick, securing a 1–0 victory for his club.[3][4]
Cheberko transferred to Zorya Luhansk on 24 June 2017, following FC Dnipo's relegation from the Ukrainian Premier League. The transfer allowed him to continue competing in the top flight of Ukrainian football, and he agreed to a two-year contract with his new club.[5]
On 5 February 2021, Cheberko joined Croatian First Football League side Osijek on an 18-month loan deal with a purchase option. The option was triggered on 2 January 2022 making the loan move permanent.[9][10]
On 9 June 2023, Cheberko transferred from Osijek to Columbus Crew.[11]