Diego Simeone made 10 changes from the 3-0 Copa del Rey defeat of Lleida, with Diego Costa among those left on the bench.
Atletico looked capable of making light work for Eibar early on, playing some great football and creating decent chances for Angel Correa and Koke.
As such, it came as little surprise when the breakthrough did arrive, as Gameiro rounded off a fine break with his fourth league goal of the season.
Thereafter the visitor offered very little from an attacking perspective, an aspect of its performance which stayed with it deep into the second half, with Eibar looking more purposeful.
Eibar's need for an equaliser saw Atletico defend desperately towards the end and the host also had a penalty shout turned down, but Simeone's men held on to move to within six points of pacesetter Barcelona, which heads to Real Sociedad on Monday (NZDT).
Atletico looked unperturbed by Eibar's excellent recent run of form during the early exchanges, as it took charge and crafted a couple of excellent chances.
The first came about at the end of a well-worked passing move by the visitors, Correa sending a cross-cum-shot – with Gameiro almost getting to it – agonisingly wide of the bottom-left corner.
And a similar move in the 11th minute presented Koke with the chance to score, but the Spain international slammed into the side-netting following Sime Vrsaljko's cross.
There was to be no denying Atleti just before the half-hour mark, though.
Antoine Griezmann sprang the offside trap when chasing Saul's through pass and coolly squared to his compatriot Gameiro, who tucked past the helpless Marko Dmitrovic.
Eibar nearly responded with a quick equaliser, but Sergi Enrich headed Joan Jordan's right-wing cross just over.
The home side offered a lot more attacking intent after the break and enjoyed plenty of possession, with the half-time introduction of tricky Chile international Fabian Orellana making a significant difference.
Even with its improved performance, Eibar had to rely on a slice of luck for its first clear chance of the second half just after the hour mark, as Enrich's header took a huge deflection off Diego Godin and was crucially stopped on the line by a relieved Jan Oblak.
Eibar's pressing almost resulted in them getting punished on the break 16 minutes from the end, but Dmitrovic did well to block Griezmann's effort at the end of a swift counter.
Atletico somehow survived in stoppage time, as Kike Garcia sent a glancing header just past the left-hand post, before Augusto Fernandez got away with handling the ball in the area in the build-up to another brilliant chance which Enrich prodded wide from close range.