The blast destroyed the doors of the office and caused significant damage to the interior. A nearby parked car was also badly damaged.
Javier Balza, the regional interior ministry spokesman, said that police had evidence concerning the culprits: "There is an idea, or analysis, shared by local, regional and national police forces about the structure of what we are facing."
The Socialist party condemned the blast as a "terrorist attack".
Alfredo Perez Rubalcaba, the interior minister, said Eta "knows it has been defeated, it's up [to the organisation] to decide when to put an end to the violence".
A bomb also exploded on Thursday at a Socialist party office in Bilbao, the Basque region's main city, slightly injuring seven policemen as they attempted to evacuate the area.
Elgoibar is an industrial town on the banks of the Deba River, 13km from Spain's northern coast.
More than 800 people have been killed by the Eta since the late 1960s. The group wants to create a separate state in the Basque region.