On 15 February 2022, Blaswich signed a contract with RB Leipzig, effective from 1 July 2022 to 30 June 2025.[6] On 20 August, he made his debut for Leipzig in a 1–2 away loss against Union Berlin.[7] On 5 October 2022, during a Champions League matchup against Celtic, Leipzig's longtime keeper Péter Gulácsi suffered a torn ACL rupture, which would force him to miss the rest of the season, in which Blaswich immediately replaced him in the 13 minutes of the first half, in a match which ended in a 3–1 victory.[8] Three days later, he played a full 90-minute matchup as a replacement in a 1–1 away draw against Mainz 05.[9]
He later managed to achieve his first titles at the club, by winning the DFB-Pokal and DFL-Supercup. On 1 September 2023, he extended his contract with RB Leipzig until 2026.[10]