Visitors Athletic led through Oscar de Marcos' first-time finish at the back post from Markel Susaeta's cross.
Lionel Messi appeared as a second-half substitute and struck the post as well as going close with a free-kick.
But sub Munir El Haddadi came to the rescue for the hosts, poking in a late equaliser from Messi's low cross.
Brazil midfielder Phillipe Coutinho also struck a shot onto the crossbar and Luis Suarez was unable to convert the follow-up.
Ernesto Valverde's side have dropped seven points in a week after being held by Girona last weekend and suffering a shock defeat by Leganes on Wednesday.
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