La Liga leaders Barca had won 10 successive games in all competitions, scoring 31 goals in the last seven, but they ran out of ideas against their resolute Basque visitors.
Valencia dominated against local rivals Villarreal but were held 0-0, while second division Cordoba also drew after coming back to make it 1-1 against Deportivo La Coruna.
Pepe Diaz pounced on an error to equalise late on after Depor had taken the lead with a Riki penalty.
The other first legs are to be played on Wednesday when Real Madrid host Levante and holders Sevilla entertain Malaga.
The second legs are scheduled for January 5.
Barca and Bilbao, who have won the cup 48 times between them, last met in the competition in the 2009 final when Pep Guardiola's side ran out 4-1 winners on their way to a six-trophy haul that year.
PRETTY PATERNS
The hosts started with the likes of Lionel Messi, David Villa and Sergio Busquets on the bench, but for all their possession and pretty patterns they lacked a cutting edge.
Gorka Iraizoz tipped Andres Iniesta's effort over the bar in the first half, but it was not until Villa and Messi joined the game after the break that the pressure grew on his goal.
Villa had the ball in the net near the end, only for the effort to be ruled out for off-side, and Gerard Pique skewed a good chance wide.
"Some days the ball just doesn't want to go in," Barca's Brazilian defender Dani Alves told reporters.
"We'll have to make a great effort because in San Mames it will be tough"
At Mestalla, Valencia controlled the east-coast derby but the closest they came to scoring was when Juan Mata volleyed against the post just after the re-start.
Villarreal, third in La Liga, slowly came into the game and Vicente Guaita denied Nilmar in a one-one-one near the end.
