Prior to the defeat, the hosts had not tasted defeat at the Estadio Manuel Martinez Valero in all competitions since December last year.
However, first-half goals from Dani Garcia and Raul Albentosa brought about an end to that run as La Liga newboys Eibar picked up their second win since returning to the top flight.
Garcia got the visitors off to an ideal start when his long-range strike opened the scoring after just three minutes.
The former Real Sociedad midfielder saw his effort fly in with a nick off the post before Ferran Corominas missed an opportunity to level shortly before the half-hour mark.
Corominas failed to convert from inside the area and Raul Albentosa made the hosts pay when he doubled Eibar's advantage three minutes before the break.
The defender headed home to give the visitors a comfortable half-time lead and, while Fran Escriba's men improved after the break, it proved in vain.
Gaizka Garitano's side held firm to pick up three points and move into fifth ahead of this weekend's remaining action.