Gaveaux was born in Béziers and sang in the cathedral choir there from the age of seven. Although intending to enter the priesthood, he also took lessons in composition. He next became first tenor at the Basilica of Saint-Seurin in Bordeaux, studying with Franz Beck, and subsequently decided to follow a career in music, becoming a conductor at the Grand Théâtre de Bordeaux as well as continuing to sing.
After a period in Montpellier, he moved to Paris where, on 26 January 1789, he took part in a performance of Giacomo Tritto's Le Avventure Amorose, which marked the inauguration of the Théâtre de Monsieur company in the Salle des Machines at the Tuileries Palace.
He subsequently performed with the company in operas such as Paisiello's L’Infante de Zamora (in 1789), and on 18 July 1791 he sang the role of Floresky in the première of Cherubini's Lodoïska. When the company moved to the Théâtre Feydeau, he was involved in a "folly in verse" called Le club des bonnes gens which was banned by the censor for being unpatriotic.
He was active during the revolutionary period, composing in 1792 a hymn to the Supreme Being. On 19 January 1795, his famous Jacobin song Le Réveil du peuple (The People's Dream), to words by Jean-Marie Souriguière de Saint-Marc, was first performed. Notwithstanding the banning of the song on 8 January 1796 by the Directoire, he continued his career in opera, appearing in François Devienne's Les visitandines, and creating the role of Jason in Cherubini's Médée on 13 March 1797.
Gaveaux's first opera, L'amour filial (1792), was a success in Paris and was performed throughout Europe: Brussels, Cologne and Rotterdam in 1795, Berne and Moscow in 1809, Berlin and Hamburg (in a German translation) in 1796.
His most famous opera, Léonore, ou L’amour conjugal, premièred in 1798, with Gaveaux himself in the role of Florestan and Julie-Angélique Scio as Léonore. It is best-known today because the libretto (by Jean-Nicolas Bouilly) served as the basis for Beethoven's only opera, Fidelio.
Other operas by Gaveaux that were popular in their day include Sophie et Moncars, Le bouffe et le tailleur and Monsieur Des Chalumeaux, and in 1808 he composed L'échelle de soie to a translation of the libretto which Giuseppe Maria Foppa had written for Rossini's La scala di seta.
Gaveaux continued to sing until 1812, although after the company of the Théâtre Feydeau merged with that of the Théâtre Favart in 1801, his voice was in decline and he only performed secondary roles. In 1819 he entered the asylum at Charenton on the outskirts of Paris, where he died. His wife, Émilie Gavaudan (also a singer), died in 1840.