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.