Anton Klementyev was born on March 25, 1990 in Togliatti, Russia. Klementyev was selected by the New York Islanders in the fifth round of the 2009 NHL Entry Draft and joined the club's AHL affiliate, the Bridgeport Sound Tigers, for the 2009-10 season. He would also get the chance to make his NHL debut during the 2009-10 season, appearing in a single game for the Islanders.
Klementyev was part of the gold medal-winning Russian team at the 2010 World Junior Ice Hockey Championships held in Buffalo, New York.