A year after earning his degree in classical piano and musicology in 1977, he met Jacques Thollot's quartet bassist Jean-Paul Céléa, with whom he formed a duo.
Over time, he played with French jazz musicians such as André Ceccarelli, Eddy Louiss, Michel Portal, François Jeanneau and with Daniel Humair.
After first meeting during the 1985 edition of the Festival de Carthage, Couturier collaborated on Tunisian oud player Anouar Brahem's albums Khomsa (1994), Le pas du Chat Noir (2001) and Le Voyage de Sahar (2005), and the two have toured together several times since 2001.
In 2006, his album Nostalghia – A Song for Tarkovsky was released on the label ECM Records alongside soprano saxophonist Jean-Marc Larché, accordionist Jean-Louis Matinier and cellist Anja Lechner.
In 2007, he played in the quintet "Passaggio", with which he recorded two albums on Label Bleu, and in the "European Jazz Trio" with Céléa and Wolfgang Reisinger.