Eibar appeared to have snatched a shock victory when Sergi Enrich finished in the 87th minute, only for Borja to salvage Atletico a point with virtually the last kick of the game.
But there were no celebrations at the final whistle. Atletico were considered serious title challengers this season but another two points dropped means they have now taken only five from their first four games.
Real Madrid and Barcelona, who both play later on Saturday, could be seven points clear by the end of the day.
Atleti created enough chances. They had 18 shots in total, but Antoine Griezmann and Diego Costa were both wasteful and as the game wore on, Eibar's confidence grew.
Diego Simeone might have liked to keep one eye on their Champions League opener against Monaco on Tuesday, with this year's competition holding extra kudos for the club, because the final will be held here at the Wanda Metropolitano next June.
But defeat to Celta Vigo two weeks ago left them five points adrift of Madrid and Barca, and while the season is still young, a victory here was much-needed to kickstart their campaign.
Costa was available, after the striker missed Spain's two internationals for personal reasons, while summer signings, Thomas Lemar and Rodrigo, both came in.
But Lemar, who was bought for 70million euros from Monaco, and should play against his former club in midweek, is yet to find his feet at Atleti and the Frenchman came off just before the hour-mark after a disjointed display, to muted applauds.
Eibar could have taken the lead when a sizzling half-volley from Cote grazed the crossbar in the opening five minutes but that was close as they came in the first half, which was largely an impressive showreel of Atletico misses.
Griezmann was guiltiest, first seeing a shot saved by Marko Dmitrovic with his feet and then failing to tee up Costa when he would have been better finishing himself.
Saul, Diego Godin and Costa were all too imprecise to beat the imperious Dmitrovic but Koke had no excuse when he too feathered wide from six yards shortly after half-time.
After yet another missed pass, Lemar was replaced by Angel Correa but as Atletico became frantic, Eibar could have capitalised, Anaitz Arbilla's drive again clipping the crossbar on its way over.
With 20 minutes left, Griezmann raced clear but lashed the ball into Dmitrovic's body.
It seemed the game could even end in defeat for Atletico when the ball landed perfectly for Enrich to smash in for Eibar from two yards.
Borja, however, spared some of Atletico's blushes with a superb shot into the corner deep into injury-time but the roar from the crowd was short-lived.
There was only disappointment at the final whistle.