After the murder of Johann Joachim Winckelmann in 1768, Visconti succeeded him as the Papal States' Superintendent of Antiquities (Commissario delle Antichita), a post which he retained until his death in 1784.
His main task was to reorganise and secure new acquisitions for the Vatican Museums and he became the first curator of the Museo Pio-Clementino, commissioning its neoclassical form.
He also controlled the granting of export-licences to archaeologists and dealers throughout the Papal States - such as Gavin Hamilton and Thomas Jenkins.
A first volume of his catalogue of the Vatican collections appeared under his name in 1782; which was continued after his death by his son, Ennio Quirino Visconti.
After his death his son, Filippo Aurelio, succeeded his father as Commissario and in 1785 was made director of the Musei Capitolini.