The Spanish team appeared to have sealed their place in the semi-finals as they led Borussia Dortmund 2-1 with seconds remaining at the Westfalenstadion, but two goals in stoppage time saw the Germans secure a 3-2 win to go through.
Malaga, playing in the competition for the first time, were fuming with some of the decisions made by Scottish referee Craig Thomson and his team, and announced that they would be making a formal complaint to UEFA about the officiating.
The Primera Division outfit were particularly upset with Felipe Santana's 93rd-minute winner which he scored from an offside position.
Al Thani, meanwhile, launched a furious Twitter outburst immediately after the match, claiming his club had been the victims of "racism", and he continued on Wednesday.
"Yes, we were targeted from the beginning of the season by corrupt UEFA and based on racism," he wrote on his personal account, @ANAALThani.
In December, Malaga accused UEFA of unfair and unjust treatment after they were handed a one-season ban from European competition - to be imposed if the club qualify for the Champions League or Europa League during any of the next four seasons - due to outstanding debts.
UEFA general secretary Gianni Infantino said the comments would now be looked at by the governing body.
He said: "It will now be analysed by our disciplinary inspectors, they will act if they have to act depending on what has been said.
"I can understand when you lose a match in the 93rd minute the emotions come up and maybe you say things that you don't really think and you really don't want to say."
- Sports & Recreation