Asier Garitano's men were consigned to a second successive defeat by Alejandro Galvez's second-half header.
Victory for Eibar was their first since the opening day of the season and helped them mark the 70th anniversary of their Municipal de Ipurua ground being opened in style.
The visitors had the better chances in a quiet first half, with Omar Ramos denied by Marko Dmitrovic before Eibar's goalkeeper produced a good save to deny Naranjo in first-half stoppage-time.
But the home side took the lead in the 53rd minute when Galvez found Takashi Inui on the left and then raced into the area to get on the end of the winger's cross and head home his first goal for the club.
Inui fired wide as the hosts looked for a second but Galvez's goal proved enough.
