Yablonsky entered Moscow's Central School of Music at age six, and made his concert debut as a cellist at age nine. He immigrated with this family to the United States in 1977, studied at Curtis Institute of Music and at Juilliard, and ultimately took his degree from Yale University. This led to an extensive career as a concert virtuoso that took Yablonsky around the world.
By chance, Yablonsky happened to be performing at a festival in Camerino, Italy, when he was asked to fill in at the last minute as conductor in the Stravinsky Octuor with the wind players from the Santa Cecilia Orchestra. Since then, Yablonsky's cello has taken a backseat to a flourishing conducting career, which took him around the world once again, however this time the carousel has stopped in his hometown, as he was ultimately named principal guest conductor of the Moscow Philharmonic. As a conductor, Yablonsky has led a variety of Russian and Eastern European orchestras for the Naxos and Marco Polo labels in mostly Russian, Eastern European, and Asian classical literature. Discs of Yablonsky's cello playing are somewhat harder to come by, but they have appeared on the Sonora Productions, Koch Discover, and Chandos labels. Yablonsky's recording for Erato of trios by Tchaikovsky and Shostakovich, with violinist Ilya Repin and pianist Boris Berezovsky, has been singled out for particular praise by critics.