Simeone's defense conceded six goals in the last eight games, but the team's fragility on set pieces - one of the areas in which it used to thrive - has also caused concern.
Atletico's fate remains in their own hands, but they need to be a lot more clinical if Costa is to avoid the ignominy of making his European return in the new year in the Europa League as his old club progress to the Champions League last 16.
"Explanations are excuses. We didn't start badly, we could've done more damage but weren't precise and Qarabag scored", Simeone said during his post-match news conference. "If they play the way they did in Madrid against Atletico, yes they can win it". Their goalkeeper had a great match and Qarabag defended very well, ' he said. Thomas has arguably been Atletico's best player this season and it was a just reward for his efforts, even though he was replaced soon afterwards by Nicolas Gaitan in the 62nd minute. We are disappointed, we have to be self-critical because we have not been able to win any of the two matches at Qarabag'.
"I believe you get what you deserve", said Simeone.
However, given that Man United are the guests of honour at Stamford Bridge on Sunday, Chelsea players can expect a few hours of defensive drills this week.
The La Liga giants have managed just two points from their opening two Group C encounters, leaving them three points adrift of second-placed Roma, who are in action against leaders Chelsea.
Having won the Greek Superleague three times, completing the domestic double twice, across two spells in charge of the home team, Valverde was warmly welcomed back to the Karaiskakis Stadium. "Now things have changed, we concede more goals and no matter how many chances we create we can't seem to score", said captain Gabi.
Neymar missed a glorious opportunity to open the scoring in the first minute before Marco Verratti's first goal of the season after 30 minutes broke the deadlock.
Atletico Madrid have reached the final in 2014 and 2016. This could certainly work in our favour. His style, personality, approach and the circumstances surrounding his prosperity are impossible to emulate, but that will not prevent the majority from trying.
As the individual who presided over the most remarkable period of sustained success in modern football history, Sir Alex Ferguson can probably be considered something of an oracle when it comes to management.
While in past years Diego Simeone's side out-muscled continental heavy weights such as Barcelona, Bayern Munich and Juventus, this campaign they have been unable to beat Azerbaijan's little-known Qarabag in two attempts.
"When you score people talk about your strength, your courage".