Son of the famous engraver, Cochin learned the art from his father and went on to complete his training with two artists of the caliber of Le Bas and Restout. His reputation led him to work for Louis XV, reproducing many public and political occasions linked to life at court in his engravings. When Madame de Pompadour sent her brother, the marquis de Vandières et Marigny, to Italy in 1749 as superintendent of the royal residences, Cochin accompanied him on the journey, along with the architect Soufflot and the abbé Le Blanc.

On his return home, Cochin published the work Voyage pittoresque d'Italie (Jombert, Paris 1758), which soon became a useful guide for travelers in Italy, thanks in part to its interesting illustrations. No less important was the publication of his essay Observations sur les antiquités de la ville d'Herculanum which contributed, along with Caylus's work, to the spread of the vogue for the "Greek" style in France.

In 1755 Cochin became secretary and historiographer of the academy, but resigned from the post a few years later to take up that of counselor, in order to leave himself more free time for his humanistic researches. With the outbreak of the French Revolution and the ensuing political changes in the country, Cochin fell into disgrace and died a few years afterward.