Morozov is a Lokomotiv Moscow academy graduate. From 2019, he played in the Youth Football League (MFL), scoring 6 goals in 34 matches, and for Lokomotiv's reserve team Kazanka Moscow, scoring 4 goals in 21 matches.
On 27 May 2022, Morozov joined Fakel Voronezh on loan for the 2022–23 season.[3] He made his Russian Premier League debut for Fakel on 12 August 2022 against Ural Yekaterinburg[4] and later become a strong first-team player. He scored his first goal for Fakel on 11 September 2022 in a 2–0 home win against Lokomotiv Moscow, the club to which he was contracted to at the time.
On 22 August 2023, Morozov extended his contract with Lokomotiv to June 2026.[5]