The result has left Atletico on two points in Group C, and the Madrid club now face a serious battle to make the knockout round of the competition.
A bright start to the match from Qarabag saw Dino Ndlovu cause Atletico's back four problems with his pace and movement, and the visitors struggled to settle inside the atmospheric stadium.
Atletico, on the other hand, were really struggling in the final third of the field, with Antoine Griezmann, Kevin Gameiro and Yannick Ferreira Carrasco finding it difficult to influence proceedings in Baku.
Two minutes later, Carrasco had a brilliant opportunity after driving into the Qarabag box, but the Belgium international could not find a route past the home side's goalkeeper Ibrahim Sehic.
Madatov had a decent strike deflected wide of the Atletico post 10 minutes before the end of the first period, before Saul Niguez headed into the arms of Sehic down the other end, but the best two chances of the first half came in the final minutes.
Griezmann had the first after dancing into the Qarabag box, but Sehic was down to make a fine save. Jose Gimenez was next to come close from the resulting corner, with the centre-back firing wide of the post from close range.
Griezmann had the ball in the back of the Qarabag net in the 53rd minute, but the offside flag denied the Frenchman a goal in the early stages of the second period.
Almeida had the home side's first attempt of the second half just past the hour, but the midfielder's effort was straight down the throat of Atletico goalkeeper Jan Oblak, before Pedro Henrique sliced wide from a good position moments later.
Qarabag were reduced to 10 men in the 75th minute when Ndlovu picked up a second yellow card for a perceived dive inside the Atletico box, which handed a big advantage to the visitors ahead of the final stages.
Torres headed just wide of the Qarabag post five minutes from time as Atletico passed up a massive chance, before Tarik Elyounoussi came close for the home side down the other end.
Qarabag actually finished the game with nine men as Badavi Huseynov was stretchered off the field, but the home side held on to secure their first ever point in the group stage of the Champions League.