Franz Anton was a son of Ferdinand Bonaventure Graf von Harrach (1637-1706), confidant of Emperor Leopold I, and his wife, Countess Johanna Theresa von Lamberg, member of an old House of Lamberg.

Franz Anton grew up in Madrid, studied canon and civil law at the Pontificium Collegium Germanicum et Hungaricum de Urbe in Rome, and was raised in 1706 by Joseph I ad personam to the rank of Fürst.

Emperor Leopold I appointed him coadjutor of Vienna in July 1701, the Papal confirmation taking place on 1 December.

On 19 October 1705, he was appointed coadjutor of the Prince-Archbishop of Salzburg, Johann Ernst von Thun und Hohenstein, and confirmed in April 1706 by the Pope.

After the death of Thun und Hohenstein, he was on 29 May 1709 made reigning Prince-Archbishop of Salzburg.