Farina later studied ecclesiastical history at the Pontifical Gregorian University in Rome, and received his doctorate in 1965. For the following three years, he worked on his specialization at the German Foundation "Humboldt" in Freiburg and Bonn. From 1968 to 1972, he was Professor of History of the Church (covering the origin to the fall of the Western Roman Empire in 476 A.D.) and of Methodology in the Theological Faculty of the Pontifical Salesian University; he then served as dean of the same faculty until 1974, and was rector of the university for two terms (1977-1983, 1992-1997).
Farina was named regolatore of the XXI general chapter of his congregation in 1978. From 1982 to 1986, he was Secretary of the Pontifical Committee of Historical Science, and was made Undersecretary of Pontifical Council for Culture within the Roman Curia in 1986. Named Prefect of the Vatican Library by Pope John Paul II on 25 May, 1997, he was the second-highest official in that Curial office, successively under Luigi Cardinal Poggi and Jean-Louis Cardinal Tauran.
On 15 November, 2006, Farina was appointed Titular Bishop of Opitergium by Pope Benedict XVI. He received his episcopal consecration on the following December 16 from Tarcisio Cardinal Bertone, a fellow Salesian, with James Cardinal Stafford and Cardinal Tauran serving as co-consecrators, in St. Peter's Basilica. Farina was later named Archivist of the Vatican Secret Archives and Librarian of the Vatican Library on June 25, 2007, replacing Cardinal Tauran, who was made President of the Pontifical Council for Interreligious Dialogue. His titular see was elevated pro hac vice to the level of an archdiocese on the same date as well.
Pope Benedict created him Cardinal-Deacon of S. Giovanni della Pigna in the consistory of 24 November of that same year. Farina will be eligible to participate in any future papal conclaves until he reaches the age of eighty on 24 September, 2013.
On 12 June 2008 in addition to his main duties he was appointed by Benedict as a member of congregations in the Roman Curia These are: the Congregation for the Causes of Saints, Congregation for Catholic Education and the Pontifical Commission for the Cultural Heritage of the Church.