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.
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.
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.
"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"
