Liverpool stormed into a two-goal lead in a match that felt more like a critical knockout tie than a group stage affair, with Salah completing a magnificent dribble with a deflected drive past Jan Oblak (8) to score for the ninth consecutive game, before Naby Keita volleyed home from a half-clearance.
Griezmann justified his decision ahead of Luis Suarez, who shockingly started on the bench, with an early double of his own.
After a corner (20), the Frenchman tapped in from close range and then cannoned in his second (34) from the edge of the box, with Atletico benefiting from Keita's sloppy marking in the build-up to both goals.
The equalizer elicited loud celebrations from Atletico manager Diego Simeone, and things could have been even worse for the visitors at half-time, with Alisson making three excellent saves to deny Griezmann, Joao Felix, and Thomas Lemar during a breathless first half.
HOW THE GAME WENT DOWN:
Salah's incredible run of form continued with a well-taken goal after only eight minutes. Geoffrey Kondogbia deflected the ball past the hapless Jan Oblak after the Egyptian cut inside past three Atletico players and fired a drive towards goal.
After Felipe failed to clear a basic cross, Naby Keita extended the lead five minutes later with a superb strike from the edge of the Atletico box.
However, just before the 20-minute mark, the hosts drew one back. Even though Thomas Lemar was offside and prevented Joel Matip from blocking it, an Atletico corner broke to Koke on the edge of the Liverpool box, and his shot was deflected in by Antoine Griezmann.
Soon after, the former Barcelona star had a fantastic chance to equalize when Kieran Trippier put him one-on-one with Alisson, but the Brazilian rose to the occasion to deny him.
Griezmann, on the other hand, didn't have to wait long to find the equalizer. Joao Felix surged into the channels, attracting Matip and Trent Alexander-Arnold's attention, while his strike partner pushed the ball into the vacated space and struck the far corner.
Before the break, Felix and Lemar both had chances to dramatically turn the game around, but Alisson was there to stop them both.
At the start of the second half, Oblak went all out to keep out a Sadio Mane header, before Alisson came to Liverpool's rescue with a close-range save from Yannick Carrasco.
However, with Atletico in command, Griezmann was sent off for a high challenge on Roberto Firmino, striking the Brazilian in the face while attempting to control the ball.
As Liverpool attempted to take advantage of their one-man advantage, Virgil van Dijk and Andy Robertson both went close, and the Reds were awarded a penalty after Mario Hermoso collided with Diogo Jota.
Salah scored from the spot-kick to give the visitors the lead again, but Atletico soon felt they had their own penalty when Jota tripped Jose Gimenez.
Regrettably, following a lengthy VAR review, the referee reversed his decision and ruled that there was insufficient contact for the Uruguayan to be granted a spot-kick.
Deep into injury time, Angel Correa had a half-chance to score, but he couldn't keep his shot down, and Liverpool held on to victory.
Liverpool: Alisson (8), Alexander-Arnold (7), Matip (6), Van Dijk (6), Robertson (6), Keita (6), Milner (6), Henderson (7), Salah (7), Mane (6), Firmino (6).
Subs used: Fabinho (7), Jota (6), Oxlade-Chamberlain (6), Gomez (1), N.Williams (1)
Atletico Madrid: Oblak (6), Trippier (6), Felipe (7), Kondogbia (6), Hemoso (5), Carrasco (7), Lemar (7), Koke (7), De Paul (6), Joao Felix (7), Griezmann (7).
Subs used: Gimenez (6), Correa (1), Llorente (1), Lodi (1), Suarez (1).
Man Of The Match: Alisson Becker
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