Opposed to the repressive regime of Governor Juan Manuel de Rosas, Acosta emigrated to Europe shortly after his secondary school graduation.

He served as a lieutenant in the Patricios Regiment, and took part in Valentín Alsina's successful pro-autonomy uprising in 1853 against the victor at Caseros, Justo José de Urquiza.

Acosta, in 1871, also served as a legal adviser to the newly formed Western Railway, and as President of the Bank of the Province of Buenos Aires.

Governor Acosta promulgated a new provincial constitution in December 1873, and shortly afterwards, he was named running mate to the National Autonomist Party nominee, Nicolás Avellaneda.

He also served as President of the Commission for the Repatriation of the Remains of General San Martín, which Avellaneda had created in 1874, and which fulfilled its objective in 1880; the Liberator of Argentina and Peru had died in France in 1850.