Kleiber was born in Berlin, the son of the conductor Erich Kleiber. As a boy, he moved with his father to Buenos Aires, who had resigned his post at the Berlin Opera in protest over the Nazi Party's policies. He later studied music in Zürich. He was conductor of the Opera there from 1964 to 1966. Later, he held posts in Stuttgart and Munich.
He made his British debut in 1966 with a performance at the Edinburgh Festival of Alban Berg's opera Wozzeck, a work his father had given the premiere of in 1925. He later made his American debut with a performance of Giuseppe Verdi's Otello.
Kleiber has largely kept out of the public eye. He conducts quite infrequently, and has made only a small number of recordings. Almost all those that he has made, however, are regarded as very fine recordings, with his versions of Ludwig van Beethoven's fifth and seventh symphonies with the Vienna Philharmonic Orchestra being near-legendary.