The Catalan side have now lost each of their last four league matches against Real Madrid, while Koeman has lost each of his three Clasicos in charge, having also tasted defeat in both league games last term.
The Dutchman is the first head coach since Patrick O'Connell, who managed the club between 1935 and 1940, to fail to triumph in his first three games against Los Blancos.
The defeat has left Blaugrana in ninth position in Spain's top flight, six points behind leaders Real Sociedad, who have played a game more, and five behind second-placed Real Madrid.
The Barcelona supporters attacked the car of Koeman as he left the stadium following the Clasico loss. body check tags ::