Asquini was educated at the Collegio Ghislieri then the Collegio Romano (where he received a Doctorate of Theology) and finally the La Sapienza University where he studied law.

He was elected Titular Latin patriarch of Antioch in 1837 and was consecrated by Carlo Cardinal Odescalchi.

He was elevated to Cardinal in pectore on 22 January 1844 and was appointed Latin Patriarch of Constantinople, a position he held until 1851.

On 24 April, he was appointed Cardinal-Priest of St Stefano al Monte Celio where he served for almost 33 years until 1877.

Finally, in 1877 he was appointed Cardinal-Priest of San Lorenzo in Lucina where he served until his death in 1878.