Salah came up with the late winner from the penalty spot after the Reds blew an early two-goal lead, with Atletico fighting back to level before half-time via an Antoine Griezmann double.
Griezmann was sent off for a high boot, while Diogo Jota was bundled over for Liverpool’s 78th-minute penalty before the Spaniards won a spotkick of their own, only for VAR to reverse it, leaving Atleti boss Diego Simeone frustrated.
Simeone did not shake Liverpool boss Jurgen Klopp’s hand after the game, racing down the tunnel after a tense game.
“We don’t like that but yes the situation is clear,” Klopp told BT Sport.
“I want to shake his hand. His reaction for sure, like mine, was not so cool.
“The next time we see each other we will shake hands definitely. It’s nothing. He was obviously angry, not with me but with the game. There is nothing else.”
Simeone explained why he did not shake hands, with claims he snubbed Klopp.
“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.”
Elsewhere in Group B on Tuesday, AC Milan slumped to their third straight defeat, going down 1-0 to FC Porto.
In Group A, Lionel Messi scored a double as Paris Saint-Germain came from behind to win 3-2 over RB Leipzig, while Manchester City thrashed Club Brugge 5-1.
In Group C, Ajax continued their perfect record with a 4-0 rout of Borussia Dortmund, while Sporting Lisbon triumphed 4-1 at Besiktas.
In Group D, Real Madrid moved level on points with leaders Sheriff Tiraspol after a 5-0 win at Shakhtar Donetsk, while the Moldovans lost 3-1 to Inter Milan.