History-making Salah seals dramatic Liverpool win
Antoine Griezmann's brace drew Atletico level at 2-2 before the Frenchman was dismissed in the second half – the loanee becoming the first player in the competition to score two-plus goals and be sent off in the same match.
Mohamed Salah then converted from the spot to restore the Reds' lead before Simeone's side had a stoppage-time penalty overturned by VAR after Diogo Jota had allegedly shoved Jose Gimenez.
The Egypt forward's winner meant the LaLiga side conceded more than twice in a Champions League home game for the first time under Simeone, last doing so against Porto in 2009.
After the match, Simeone sprinted down the tunnel as he snubbed a full-time handshake with Klopp, but the Atleti head coach insisted there was no malice in the gesture.
"I don't normally greet after the game," he said.
"I do not like it and I think it is not healthy because there will always be someone who is not satisfied with the game.
"They have a different culture, which I do not share. If I see [Jurgen] Klopp now, I have no problem greeting him."
Klopp collected Liverpool's first win in five attempts against Atletico in the Champions League, while victory ended a five-game winless run against Spanish sides in the competition.
However, centre-back Felipe was left disappointed after the game as he explained his confusion with the overturned penalty decision, albeit the 32-year-old admitted his vision for the Griezmann sending-off was impaired.
"I don't know what the rule is but, for me, it is clear that it is a penalty," Felipe told Movistar.
"He [the referee] went to the VAR to look, then he went to go and look again – it is clear that he had doubts.
"The other way around [for Liverpool's penalty], it was a very fast decision. We have to fight our own battles, but it is a bit unfair.
"The play [by Griezmann] was for yellow and that's it because he's looking at the ball and trying to control it."