Manuel Pellegrini's side head into next Sunday's 'clasico' at the Nou Camp with 28 points from 11 games, one ahead of Barca, after Gonzalo Higuain got the only goal at the Bernabeu.
Daniel Alves gave Barca the lead just after the break at San Mames but the Basques battled back with substitute Gaizka Toquero racing through to level just past the hour mark.
Real had been whistled off the pitch in their last home game, after being knocked out of the King's Cup by third-tier Alcorcon, and the Bernabeu became restless as their team continued to dominate but without adding to the score.
"The game wasn't as tight as the scoreline suggests. On another day, playing worse, we would probably have scored more goals," Pellegrini said.
Bilbao chased shadows for most of the first half but made life difficult for Barca until Xavi played the overlapping Alves into the area and the Brazilian fired inside the far post.
LLORENTE FLICK-ON
Bilbao, who lost 4-1 to Barca in last season's King's Cup final, ran themselves into the ground to keep their out-of-sorts visitors at bay.
"We are playing as well as we usually do, creating chances, but struggling to score goals," Alves told television station La Sexta. I hope normal service will be restored from next Sunday."
Manolo Jimenez's side held on to third place with 25 points, three ahead of Deportivo La Coruna, who climbed to fourth after a hotly-disputed injury-time penalty gave them a 2-1 victory over visiting Atletico Madrid.
Mexico's Andres Guardado netted the spot kick at the end after Pablo Ibanez was penalised for bringing down Alberto Lopo.
