Sebastiano Baggio studied in Vicenza and Rome compartments Catholic theology and philosophy. He attended the Pontifical Ecclesiastical Academy and completed additional training in palaeography and librarianship. He received in 1935 the sacrament of Holy Orders and entered the diplomatic service of the Vatican.
From 1936 to 1938 he worked at the Apostolic Nunciature in Austria. Further stations were until 1946 El Salvador, Bolivia and Venezuela. From 1946 to 1948 he worked in the Vatican Secretariat of State in Rome, from 1948 to 1950 he belonged to the staff of the Apostolic Nunciature in Colombia. On June 21, 1953 he was awarded Pope Pius XII. the honorary title house Prelate of His Holiness. From 1950 to 1953 he worked for the Consistorial the Curia. Pius XII appointed him in 1953. Titular Archbishop of Ephesus and Apostolic Nuncio in Chile. He received his episcopal consecration Cardinal Giovanni Piazza Adeodato OCD. In 1959 Sebastiano Baggio Apostolic Delegate in Canada, 1964 Apostolic Nuncio in Brazil.
He took 1962 to 1965 at the Second Vatican Council in part. Pope Paul VI. took him in 1969 as a cardinal deacon with the title Diakonia Santi Angeli Custodi a Città Giardino in the College of Cardinals, and appointed him Archbishop of Cagliari. 1973 Sebastiano Baggio Prefect of the Congregation for Bishops and Cardinal Priest of the titular church of San Sebastiano Catacombs and all on 12 December 1974, a Cardinal Bishop of Velletri and Segni. In 1974 he took over as Papal envoy at the Eucharistic Congress in Quito in part, in 1975 the Brazilian Eucharistic Congress in Manaus. In addition, he provided many years the office of President of the Pontifical Commission for Latin America. Cardinal Baggio took part in the conclave of 1978 and August- October - conclave in the same year. Pope John Paul I wanted him after his election in August 1978 appointed Patriarch of Venice, but Baggio refused.
In 1984, he became president of the Pontifical Commission for Vatican City State and Cardinal Patronus of the Order of Malta. On March 25, 1985 John Paul II appointed him Camerlengo. In 1990 he resigned from all management duties because of illness. However, the Office of the Camerlengo he retained until his death. Sebastiano Baggio died on March 21, 1993 in Rome and was buried in his hometown.