Andre Navarra was born into a musical family in Biarritz, France, in 1911. He began studying cello at the age seven. Two years later, he was accepted as a student at the Toulouse Conservatory, and graduated in 1924 with first prize. He then continued his studies at the Conservatoire de Paris, studying cello with Jules Leopold-Loeb and chamber music with Charles Tournemire. He graduated two years later at age fifteen, again taking first prize.
Navarra remained in Paris for a period of self-study, and used the opportunity to meet and observe the playing of musicians such as Emanuel Feuermann, pianist Alfred Cortot, and violinist Jacques Thibaud. Navarra also developed friendships with composers Jacques Ibert, Florent Schmitt, and Arthur Honegger. Later on, he was mentored by Pablo Casals.
In 1929, at the age of eighteen, Navarra joined the Krettly Quartet, and remained with them for the next seven years. He also helped form an ensemble called the B.B.N. Trio with pianist Joseph Benvenuti and violinist René Benedetti. Two years later, he made his solo debut with Paris's Colonne Orchestra, performing Édouard Lalo's Cello Concerto in D minor. In 1933 he became principal cellist of the Paris Opéra Orchestra, in addition to continuing to appear as a soloist with various European orchestras.
Navarra slowly continued to establish his career throughout the 1930s, receiving a major boost in 1937 when he won first prize at the Vienna International Competition. However, his career was halted by World War II in 1939 when he served with the French infantry.
In 1945, André Navarra resumed his career. In 1949, he accepted a professorship at the Conservatoire de Paris as a successor to Pierre Fournier, and toured extensively in the United States, Europe, Asia, and the Soviet Union, playing with the era's great conductors. His performances included premieres of cello concertos written for him by composers including André Jolivet, and a well-received version of Elgar's Cello Concerto with Sir John Barbirolli.
In addition to his position at the Conservatoire de Paris, Navarra taught summer courses at the Accademia Musicale Chigiana Siena from 1954, fall courses in Saint-Jean-de-Luz and accepted an additional professorship at the Hochschule für Musik Detmold in 1958. He also taught in London and Vienna.