European champions Barca were cruising after early strikes from Xavi and Cesc Fabregas but they eased off and the Basques levelled with two goals in three second-half minutes from striker Imanol Agirretxe and Antoine Griezmann.
France striker Karim Benzema led Real to victory, scoring either side of a disputed Cristiano Ronaldo penalty, but it needed substitute Gonzalo Higuain to settle the match for Jose Mourinho's men.
Real have a perfect six points, the same as Valencia who were 1-0 winners at home to Atletico Madrid. Barca have four points along with Sociedad and Sevilla who grabbed a 2-2 draw at 10-man Villarreal.
Roberto Soldado powered a header in off the bar for Valencia in the 52nd minute and tops the scoring charts with Ronaldo, on four goals from two games. Villarreal had goalkeeper Diego Lopez harshly sent off after 22 minutes at the Madrigal and Alvaro Negredo saw his penalty saved but scored on the rebound to put Sevilla ahead.
Giuseppe Rossi levelled with a penalty at the other end and Marco Ruben fired Villarreal ahead in the 73rd before Alexis Ruano headed a late equaliser. Perhaps with the midweek Champions League fixture against AC Milan in mind, Barca coach Pep Guardiola left Lionel Messi, David Villa and Andres Iniesta on the bench and at first they were not missed but as the game progressed, their absence was felt.
Deccan Herald is on WhatsApp Channels| Join now for Breaking News & Editor's Picks