Costa was born in Taranto to Angelo and Maria Giuseppa née Malagisi.

Costa studied composition, piano and singing at the San Pietro a Maiella Conservatory in Naples under another uncle Carlo Costa, Paolo Serrao, and Giuseppe Martucci.

[1][2] By the age of 17, Costa had already published numerous art songs.

On his return to Naples, he became very active in the revival of the Neapolitan musical tradition.

The following year his remains were brought to Taranto and placed in a monumental tomb erected by the city.