The Rojiblancos boss sprinted down the tunnel in order to dodge his German counterpart after his side fell to a 3-2 home loss.
The rivalry between Klopp and Simeone was simmering gently before kick-off as the German insisted he hadn't been critical of Atletico's style of play back in 2020 after their infamous Anfield Champions League defeat.
Former Liverpool striker Peter Crouch said: "I respect Simeone so much as a manager but not shaking hands at the end - he lets himself down."
While Joleon Lescott added: "It's a bit cowardly in my opinion."
The tension is sure to boil over when the two sides face each other again in with Atleti needing a result at Anfield to help their qualification chances.
But where does this heated rivalry rank among some of the others in recent history? talkSPORT.com takes a look…
Jose Mourinho vs Arsene Wenger
The two icons of Premier League history could have their own list of bust-ups having produced some incredible quotes together.
Mourinho called Wenger a 'voyeur' in 2006 - someone who likes to watch other people.
Two years later the ex-Arsenal manager referred to the Special One as 'out of order, disconnected with reality and disrespectful.'
Mourinho's 'specialist in failure' insult has lasted the test of time and the animosity between them finally came to a head in 2014.
Wenger approached the Chelsea technical area to shove the Roma chief during a 2-0 defeat after becoming enraged with a foul on Alexis Sanchez.
Like Klopp and Simeone, these two also did not shake hands at full-time and Wenger later admitted to talkSPORT it was a very personal rivalry that went overboard.
Neil Warnock vs Stan Ternent
The Middlesbrough coach's long-running feud with Ternent can be traced back to an ugly clash between Sheffield United and Burnley in 2001.
Ex-Blackpool gaffer Ternent accused Warnock of sending a spy to listen to his team talk and it didn't take long for thingd to turn physical.
In his autobiography, Warnock wrote: "I had always known Stan Ternent was a d***head but when Sheffield United played Burnley in 2001 he behaved like a deranged lunatic.
"I'd told my assistant Kevin Blackwell to keep an eye on Ternent. I knew he'd be trying to put pressure on the ref. So when Ternent came round the corner, frothing at the mouth, Blackie told him to leave it out.
"That was all the encouragement Ternent needed. He launched himself at Blackie and butted him. Blackwell swung a right hook and smacked him on the nose. He sploshed him good and proper.
"In Ternent's autobiography he tells how he gave Blackwell a good hiding. But we saw the incident differently. Blackie had a little cut on his lip. Ternent was in bits."
Neil Lennon vs Ally McCoist
The pair's infamous Old Firm bust-up lives long in the memory with their collision even leading to a Scottish government summit.
The tension between Celtic and Rangers was at an all-time high in 2011 with the Gers on the verge of going into liquidation.
And matters weren't helped when Rangers had three players sent off during a 1-0 defeat to their arch rivals leading Lennon and McCoist to get involved.
As the two managers met to shake hands, it quickly descended into a memorable head to head that caused government officials to discuss the incident in the Scottish parliament.
Lennon told the BBC in 2015: "A lot had gone on that night. There were things I didn't like, there were things Ally didn't like. We had a few words and that was the end of it.
"It made great television and it made great storylines for a long, long time.
"In 20 years, they'll be showing footage of me and Ally squaring up to each other."
Domenico di Carlo vs Silvio Baldini
Baldini was renowned for his heated temper that was very much in full show when his side Catania faced Di Carlo's Parma in 2007.
The Italian, who had been sacked from The Crusaders three years earlier, allowed the ill-fated reunion to get to him and he was sent off for dissent.
But Baldini and refused to leave and then-Parma boss Di Carlo made his presence known as the pair shared verbal insults.
And the resulting bust-up led to Baldini kicking his rival up the backside from behind and earning himself a one-month ban from the Italian FA.
As reported by the Guardian, his 'apology' read: "I apologise to everyone. To Catania's fans, to the whole city…and to Parma's fans, as my attack was certainly not intended at them.
"But I do not apologise to him (Di Carlo). I don't owe Di Carlo anything: he is a flawed person and he provoked me."
Frank Lampard vs Marcelo Bielsa
The 'Spygate scandal' is a story which will be etched long in the history of the Championship.
Leeds were fined £200,000 by the FA after Bielsa sent a member of his coaching staff to spy on a Derby training session while Lampard was in charge of the Rams during the 2018/19 season.
Derby had the last laugh however when they beat Bielsa's men in the play-offs semi-finals at Elland Road.
"I didn't call him to apologise," Bielsa claimed after the incident. "He said 'you violated fair play rules'. It was not convenient for me to explain to him my point of view."
Sir Alex Ferguson vs Kevin Keegan
Newcastle had just beaten Leeds in April 1996 and the title race between the Toon and Fergie's Manchester United was very much in its final stretch.
It should never have got to that stage as the Magpies had a 12-point lead in January but poor form let the Red Devils catch up and take charge.
And Keegan couldn't contain his emotions as he was quizzed on comments Ferguson made in his post-match interview.
United legend Fergie publicly took issue with Newcastle taking part in a testimonial match for Stuart Pearce after the season ended.
He suggested Pearce's club at the time, Nottingham Forest, wouldn't be at their best when the two sides met in the Premier League a week earlier.
Keegan reacted to Sky Sports: "We're playing Notts Forest on Thursday and he objected to that! Now that [the testimonial] was fixed up months ago…
"That sort of stuff, we're bigger than that… when you do that with footballers like he said about Leeds and when you do things like that about a man like Stuart Pearce, I've kept really quiet but he really went down in my estimation when he said that, we have not resorted to that.
"But I'll tell him, you can tell him, he'll be watching. We're still fighting for this title and he's got to go to Middlesbrough and get something and I'll tell you, honestly, I will love it if we beat them. LOVE IT!"
Neil Warnock vs Nuno Espirito Santo
Now onto the legends who made this list twice…
Warnock was left fuming once more after his Cardiff side missed TWO stoppage time penalties in a 1-0 defeat to promotion-rivals Wolves in 2018.
And the 72-year-old vented his anger at now-Spurs manager Nuno for not shaking his hand at the final whistle - eventually telling him to 'f*** off'.
In his post-match presser, Warnock said: "I don't accept it at all, it's totally out of order and I think he's a disgrace.
"I used strong words and I meant every one of them.
"He's got to learn that in British football you have manners, a bit of class when you've won a game. He's got to learn that."
Arsene Wenger vs Alan Pardew
The Frenchman couldn't help but lose his customary composure to get in a scuffle with the then West Ham coach in 2006.
The two had to be separated after Marlon Harewood's late winner for the Irons in which Alexander Hleb was denied a penalty minutes earlier.
Wenger appeared to swear at Pardew at one point before shoving the Englishman as Robin van Persie was also struck on the head by a coin.
And to the surprise of no one, Wenger refused to shake his rival's hand at the end of the match and left Upton Park without doing interviews.
Jose Mourinho vs Pep Guardiola
The two titans of the English game might never do battle in the same league again after almost a decade of being at war.
Neither Mourinho nor Guardiola are known for having physical fisticuffs but what Mourinho did during the second leg of the 2011 Supercopa de Espana overstepped the mark.
Marcelo was sent off for a scissor tackle on Cesc Fabregas and it sparked a melee on the pitch and in the technical area.
In that dramatic turn of events, Mourinho went up to Tito Vilanova, Guardiola's assistant at the time, and gouged his eye.
Mourinho was given a two-game ban while Vilanova received a one-game ban for his part in the incident.
Barcelona went on to win the trophy with a 5-4 victory on aggregate but talkSPORT has also ranked some of the duo's other iconic meetings.
Sir Alex Ferguson vs Arsene Wenger
A rivalry that defined the early Premier League and provided stories for the ages when Man United went toe-to-toe with Arsenal year after year.
Both teams were evenly matched after that and only the finest of margins separated them in league and cup competitions, with the Gunners a missed Dennis Bergkamp penalty away from a place in the 1999 FA Cup final, which would have ended United's treble dream.
Other notable incidents include Phil Neville receiving a yellow card after just 27 SECONDS in the 2003 Community Shield for crunching Vieira.
In 2003, six Arsenal players and two United stars were charged with improper conduct following the 'Battle of Old Trafford,' with Arsenal charged for 'failing to ensure the proper behaviour of their players'.
Ringleader on the day, Martin Keown revealed even his wife had told him he'd gone too far in his confrontation with Ruud van Nistelrooy.
A year, Man United ended Arsenal's 49-game unbeaten run in a match that became known as the 'Battle of the Buffet'. Furious players clashed in the tunnel afterwards culminating in Cesc Fabregas throwing a slice of pizza at Ferguson's face.
"It was a time where the Arsenal players were Arsenal players for life and the Man United players were Man United players for life," Arsene Wenger recalled to talkSPORT last year.