Kossenko was born in 1977 in Nice, France. He attended the Conservatoire de Paris, where he studied modern flute with Alain Marion and earned a first prize. He went on to the Sweelinck Conservatory in Amsterdam, where he studied with historical flute specialist Marten Root. Kossenko won several prizes performing on a modern flute, including a first prize in 1995 in the Lions Club International Flute Competition and a special prize in 2000 at the Jean-Pierre Rampal Flute Competition. However, most of his performing career has been devoted to Baroque music, and he has found himself much in demand from chamber orchestras in France and elsewhere. Among the ensembles with which he has performed are Le Concert d'Astrée, Le Concert Spirituel, and John Eliot Gardiner's Orchestre Revolutionnaire et Romantique. Kossenko has also made an impact as a conductor, leading such ensembles as B'Rock, Arte dei Suonatori, and the Holland Baroque Society. Kossenko has played chamber works with keyboardist Richard Egarr, pianist Leif Ove Andsnes, and violinist Andrew Manze, among others. He founded his own period-instrument ensemble, Les Ambassadeurs, and has often toured and recorded with that group as well as with Arte dei Suonatori.
Kossenko has made more than a dozen recordings, many of them on the Alpha label. He has also recorded for Glossa and Channel Classics, among other labels. With Les Ambassadeurs, he released the album Vivaldi: Concerti per l'Orchestra di Dresda on Alpha in 2013. In 2021, he moved to Claves for a recording of the Symphonies Concertantes of Antoine Reicha with Gli Angeli Genève. ~ James Manheim, Rovi