Carlo Ancelotti's side took a narrow lead into the second leg, before that was wiped out inside 30 seconds by Atletico midfielder Conor Gallagher.
Vinicius Junior blazed a second-half penalty high and wide, and with neither team able to win the tie in 120 minutes, the winner was decided by a penalty shoot-out.
The first three penalties were dispatched before Alvarez stepped up to take Atletico's second spot-kick in the tense shootout at the Metropolitano.
Alvarez's standing leg gave away in his kicking motion, causing him to slip as he fired his penalty past Thibaut Courtois.
UEFA to discuss double touch rule change
The Argentine thought he had scored a legitimate spot-kick until a VAR review ruled out the goal for a double touch.
Real Madrid went on to win 4-2 on penalties to set up a quarter-final clash against Arsenal, while Atletico were left feeling aggrieved following the controversial penalty call.
UEFA have since released new footage which shows Alvarez's left boot making contact with the ball before he shoots with his right.
They also confirmed in a statement that they would discuss a potential rule change will the relevant organisations following Wednesday's controversy.
UEFA said: "Atletico de Madrid enquired with UEFA over the incident, which led to the disallowance of the kick from the penalty mark taken by Julian Alvarez at the end of yesterday's UEFA Champions League match against Real Madrid.
"Although minimal, the player made contact with the ball using his standing foot before kicking it, as shown in the attached video clip. Under the current rule (Laws of the Game, Law 14.1), the VAR had to call the referee signalling that the goal should be disallowed.
"UEFA will enter discussions with FIFA and IFAB to determine whether the rule should be reviewed in cases where a double touch is clearly unintentional.
What is the current rule on double touch penalties?
The current rule states: "[A penalty] kick is completed when the ball stops moving, goes out of play or the referee stops play for any offence; the kicker may not play the ball a second time."
It now seems that UEFA will hold talks with FIFA and IFAB about whether ruling out a goal is the suitable punishment for scenarios such as the one that was seen at the Metropolitano.
They will decide whether to stick with the current rule or look a alternative solutions, potentially allowing a retake rather than chalking the goal off.
Written by
Ben Sully