Cheberko is a product of the UFK Dnipro and FC Dnipo youth systems. On 24 July 2016, Cheberko made his debut for FC Dnipro, coming on as a second half substitute in a match against FC 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. Cheberko transferred to Zorya Luhansk on June 24, 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. In July 2020, it was announced that had transferred to Austrian Bundesliga side LASK for a reported €300k 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. On 9 June 2023, Cheberko transferred from Osijek to Columbus Crew.
Club Club Season League Cup Continental Total Division Apps Goals Apps Goals Apps Goals Apps Goals Dnipro 2016–17 Ukrainian Premier League 14 2 1 0 — 15 2 Zorya Luhansk 2017–18 10 0 0 0 — 10 0 2018–19 16 0 2 1 0 0 18 1 2019–20 27 0 1 0 5 0 33 0 Total 53 0 3 1 5 0 61 1 LASK 2020–21 Austrian Bundesliga 5 0 1 0 1 0 7 0 Osijek (loan) 2020–21 Prva HNL 13 0 — — 13 0 Osijek 2021–22 19 0 3 0 3 0 25 0 2022–23 20 0 3 1 0 0 23 1 Total 52 0 6 1 3 0 61 1 Columbus Crew 2023 Major League Soccer 0 0 0 0 0 0 0 0 Career total 124 2 11 2 9 0 144 4