The Reds marched to their second successive Premier League win in extraordinary circumstances at St James' Park, where Anthony Gordon haunted his former youth club in the first half.
The ex-Everton winger capitalised on a dreadful error from Trent Alexander-Arnold, who had also controversially escaped an early sending-off for two bookable offences, having been cautioned for throwing the ball away before bringing down a charging Gordon.
While Alexander-Arnold was not given his marching orders, Virgil van Dijk was made to take the walk of shame for a last-man foul on Alexander Isak just three minutes after Gordon had slotted home through Alisson Becker's legs.
However, Eddie Howe's side struggled to make their extra man count, and Darwin Nunez - who has seldom seen the field so far this term - came off the bench to equalise late on before firing home a dramatic injury-time winner in front of the away end.
Speaking to the media in his post-game press conference, Klopp described Sunday's enthralling success with 10 men as a once-in-a-lifetime experience, saying: "I think in my 1,000 games as a coach or a manager I never had a game like this, that's the truth.
"Not that we never turned games [around], we did that, but 10 men in an atmosphere like this against an opponent like this - it's not that I can't remember, I'm pretty sure it never happened because these moments are rare and super-special.
"But I thought the boys deserved it today because with 10 men we played better and gave Newcastle a proper game. Before that it was a bit wild with, obviously, advantages for Newcastle with the 1-0. We didn't play calm enough, we felt under pressure when we wouldn't have to."
Nunez's game-changing cameo was the Uruguayan's third successive substitute cameo in the 2023-24 Premier League season, as Diogo Jota and Cody Gakpo have been preferred in the number nine role over the past few weeks.
When asked if Nunez had done enough to warrant a starting role against Aston Villa next weekend, Klopp gave little away but hinted that the 24-year-old would be under consideration, adding: "I never made a decision about the line-up the week before the game, we will see. But you can have worse arguments, I would say."
However, the Reds will go into the meeting with Unai Emery's side without the services of Van Dijk, who will be suspended for at least one game owing to his straight red card on the evening.
It remains to be seen whether Van Dijk's explosive reaction will lead to an increased punishment from the FA, but Klopp was critical of John Brooks's decision to banish the Dutchman.
"I don't think it is a red card. It's pretty much no contact, very little, and it's on the way to the ball, but what can I say?" the Liverpool boss added.
"The decision is like this, I cannot change it and I don't want it. It's just, would I give this in a training game? Definitely not. There are reasons why I am not a ref and they are."
Liverpool sit fourth in the Premier League table having taken seven points from their opening three games, with only nine-point champions Manchester City enjoying a better start.