
The dull draw leaves Betis seventh from bottom in La Liga, with eight points from nine games.
Bilbao are now in 11th place with 10 points.
Both teams look likely to battle relegation, just as last season.
Betis coach Hector Cuper, true to form, planned a defensive game and got away with it, thanks to the hosts' lack of punch in attack.
The closest that Bilbao came to breaking the deadlock was at the end, when young striker Fernando Llorente came off the bench and forced a couple of sharp saves out of Portugal World Cup keeper Ricardo.
"It wasn't easy for us tonight," said Bilbao veteran Joseba Etxeberria. "Since they hardly came out of defence, they didn't take any risks at all."
Earlier Saturday, in-form Espanyol missed the chance to go joint top, being held to a 2-2 draw at Mallorca.
Had Espanyol won in Palma, they would have gone level at the top with Real Madrid, with 19 points from nine games.
Instead, they are now joint fourth with neighbours Barcelona, both on 17 points, one point behind Valencia and Villarreal.
Mallorca stay in sixth place on 15 points.
On Sunday, leaders Real are at home to troubled Deportivo Coruna, while Villarreal are at Zaragoza and Barcelona host Almeria.
Valencia will be at a Sevilla side shocked by the departure of coach Juande Ramos for Tottenham Hotspur.
Club stalwart Manuel Jimenez, until Friday coach of the club's nursery team, will be making a difficult debut on the Sevilla bench.
DPA