Durazzo was ordained in 1618 and was appointed referendary of the Tribunals of the Apostolic Signature of Justice and of Grace in 1621. Being from a wealthy family, Durazzo was able to purchase positions as clerk of the Apostolic Chamber between 1621 and 1623 and protonotary apostolic in 1624. He became treasurer-general of the Apostolic Chamber in 1627 and held that position until 1633 before being appointed pro-secretary of the Apostolic Chamber in 1633.
In 1635 he was appointed archbishop of Genoa, a role he retained until his death. As archbishop of Genoa, Durazzo also conducted mercantile business in addition to his ecclesiastic work and was able to become very wealthy. Contemporary John Bargrave also suggested that he developed a reputation for spending vast sums of money on gambling and women.
Between 1640 and 1642 he was appointed legate in Bologna and Ferrara during the First War of Castro but left when Antonio Barberini arrived with papal forces, forcing papal representatives to return to Rome.