Barca were without the injured Lionel Messi, who is not expected back until mid-January, and turned in probably their worst performance of the campaign against a fired-up Bilbao at the Basque club's new San Mames stadium.
A miserable week for the Spanish champions, which included a 2-1 Champions League loss at Ajax Amsterdam, ended on a worse note when Bilbao's Iker Muniain turned in a Markel Susaeta cross in the 71st minute.
It was only the second time in 15 matches this season that Barca have dropped points and they are level with Atletico on 40 after the Madrid club's 2-0 win at promoted Elche on Saturday.
Real, whose 4-0 drubbing of Real Valladolid on Saturday included a hat-trick from record signing Gareth Bale, are three points further back in third.
(Reporting by Iain Rogers, editing by Ed Osmond)
