La Liga's top scorer Lionel Messi had been allowed to miss the match in order to rest ahead of the Copa America with the shock home defeat to Celta Vigo the previous weekend ending Barca's title hopes.
Neto started in goal in place of Marc-Andre ter Stegen, who opted to pursue treatment on a knee problem which also rules him out of Euro 2020, while Francisco Trincao was part of an attacking trio which also included Antoine Griezmann and Ousmane Dembele.
It was a game of few chances with Eibar looking the more likely to score in the first half as the came close on various occasions. Kike, in particular, forcing Neto into a save.
Barca lost Ilaix Moriba to injury in the second half before Ousmane Dembele came close to opening the scoring as he danced through the Eibar defence before shooting wide.
However, the goal eventually came for the visitors when Griezmann scored with a stunning acrobatic effort 10 minutes from time.
Ronald Koeman's side were guaranteed to finish the campaign in third place -- the first time since 2008 they are not in the top two -- while Eibar's relegation had already been confirmed.