He was the youngest of five children, his siblings being named Ona, Emilija, Jonas and Marija.

He then did pastoral work in Kaišiadorys until 1959, including serving as a professor and the prefect of studies and discipline at the Kaunas seminary.

However, Bishop Sladkevičius was impeded from performing his ministry by his country's Communist government, and he took up residence at Nemunėlio Radviliškis, where he was under virtual house arrest from 1963 to 1982.

He was named Apostolic Administrator ad nutum Sanctae Sedis of Kaišiadorys on 15 July 1982, and became President of the Lithuanian Episcopal Conference on 27 April 1988.

Pope John Paul II created Sladkevičius Cardinal Priest of Spirito Santo alla Ferratella in the consistory of 28 June 1988, and later Archbishop of Kaunas on 10 March 1989.