Piatti was born at via Borgo Canale, in Bergamo and died in Mozzo, 4 miles from Bergamo.
The son of a violinist, Antonio Piatti, he originally began his studies on the violin before switching to the cello. As a cellist, he studied under his great-uncle, Gaetano Zanetti, a great cellist. After two years of studying, he joined the theater orchestra, where he played for three months - for ten shillings, half of which his grandfather took. After Zanetti's death, he became a pupil at the conservatorio of Milan under Vincenzo Merighi. He made his concert debut at 15 and started touring at 16. No one doubted the young virtuoso's skill on the instrument, but he did not draw large crowds. As a result, when Piatti fell ill during an engagement, he was forced to sell his cello to cover the medical costs. Franz Liszt invited him to appear as a guest performer at one of his recitals; stunned by what the boy could do on a borrowed cello, Liszt presented him with a superb new instrument. Piatti went on to become one of the most celebrated cellists of his day, as popular for the pieces he wrote as for the robust and unsentimental way he performed them.