Having lost the Supercopa de Espana final to Real Madrid on penalties six days ago, Diego Simeone's side looked sluggish and short of inspiration in the heavy rain at Ipurua.
Burgos struck just 10 minutes into the contest to give Eibar a deserved lead, and Jose Luis Mendilibar's side held onto their advantage with relative ease in the second half despite Atleti's control of possession.
Exposito buried a precise strike in the 90th minute to give the hosts a 2-0 victory they fully deserved.
Eibar's early pressure told 10 minutes in when Burgos tapped home following Sergi Enrich's header, the goal given after a VAR check saw the initial offside decision overturned.
Joao Felix was denied by a good Marko Dmitrovic save, but only a crucial block from Santiago Arias denied Takashi Inui a second for Eibar.
Atleti were certainly second best in the first half but Angel Correa missed two clear chances to equalise, the forward unable to connect cleanly with Vitolo's cross from barely six yards out and then failing to beat Dmitrovic after Vitolo's stumble.
Simeone opted against any changes at the break and there was little difference to Atleti's level of creativity against an Eibar side happy to hold onto their advantage.
Joao Felix worked hard to create inroads but to little effect, while Alvaro Morata saw a header comfortably saved, and the hosts secured only a second win in eight league games when Exposito arrowed a fine finish beyond Jan Oblak after a corner was cleared straight to his feet.
2 - Atlético de Madrid have suffered their second defeat against Eibar in all competitions (W12 D4), the first one since April 2001 in the second tier (2-1). Rain. pic.twitter.com/Wu8KqEOPLP— OptaJose (@OptaJose) January 18, 2020
What does it mean? Atleti in danger of being cast adrift in title race
Real Madrid's 2-1 win over Sevilla earlier meant Atleti would have moved three points ahead of Julen Lopetegui's side, and back to within five of their city rivals, had they won.
However, the nature of their display will give fans little belief they can keep pace with Madrid and Barca at the top of the table even though they have 18 games still to play.
As for Eibar, they are now seven points above the drop zone.
Although his four saves were relatively straightforward, Dmitrovic deserves credit for commanding his penalty area well. Atleti's rather hopeful crosses rarely looked likely to trouble him.
No delight for Angel
Atleti created few clear-cut openings but Correa spurned the best of them, managing only to meet Vitolo's good cross with his face rather than his head and then failing to find a finish when the ball fell kindly his way.