The visitors looked like they had snatched maximum points after 87 minutes when Sergei Enrich prodded home from close range.
Substitute Garces, making his LaLiga debut, spared Diego Simeone's blushes, though, lashing in from 15 yards, just moments after Diego Godin had thundered a header against the crossbar.
Failure to adequately bounce back from defeat to Celta Vigo before the international break means that Diego Simeone's side is now seven points behind early pacesetter Barcelona.
After a sluggish start, in which Eibar's Jose Angel clipped the crossbar with a powerful half-volley, the hosts sparked into life, only to find Marko Dmitrovic in inspired form.
First, Antoine Griezmann's low effort was kept out by the Serbian before he superbly pawed away Saul Niguez's downward header and blocked Godin's close-range drive in quick succession.
Dmitrovic was in action again just after the restart, tipping over Godin’s header and repelling Diego Costa's drive within the space of a minute.
An unmarked Koke then inexplicably failed to connect with the ball just four yards out, while Eibar once again hit the woodwork, this time Anaitz Arbilla thundering a half-volley against the crossbar from 30 yards.
The visitors thought they had snatched a memorable win three minutes from time when Enrich took advantage of a deflection into his path to clip past Jan Oblak.
Atleti's hopes of a dramatic leveller looked to be over when Godin's header crashed back off the bar in stoppage-time, but 19-year-old Garces had other ideas.