The Reds extended their perfect Anfield run to 11 wins from 11 home matches in the 2023-24 season, although they were staring down the barrel of defeat until the final few moments.
A chaotic first half saw Trent Alexander-Arnold's free kick cannon off Bernd Leno into the back of the net, while Alexis Mac Allister scored a thunderbolt from 30 yards, but Harry Wilson and Kenny Tete both punished poor defending from the hosts.
Bobby Decordova-Reid appeared to win it for the visitors when he headed home at the back stick in the 80th minute, only for Wataru Endo to direct a crisp first-time strike into the top corner just five minutes later, before Alexander-Arnold's sweetly-struck winner settled a seven-goal spectacular.
Never before had Liverpool come from a goal down to win a Premier League game as late as they did on Sunday, and owing to Manchester City's 3-3 draw with Tottenham Hotspur, the Reds end the weekend sat second in the table behind Arsenal.
However, Klopp was under no illusions that his side were a tad fortunate to come away with all three points, telling the press: "The feeling after the game was exceptional, to be honest. During the game, it was slightly different in moments. I told the boys after the game, the game turned out as the game we saw because we were a bit dumb.
"We were unlucky with Darwin [Nunez] and the crossbar and other situations where we could have scored. Then they scored the 3-2 and that was not not-deserved as well, you have to say that. Football is sometimes like this.
"We changed system a couple of times and that worked out. It is not always the case, but today it worked out and then it ends up in a high-intense, hectic period we have a completely calm situation - the ball to Mo, Mo to Wata and Wata fires into the far corner, completely free. Before that it was not once the case.
"I saw a few games, but I never saw a game with this amount of worldies, to be honest. I don't think anybody would have thought before the game Liverpool v Fulham will be a game you will never forget in your life. But you're welcome. Whoever was here today will never forget it in their lives."
Once again, Klopp gave Joel Matip the nod over Ibrahima Konate in a Premier League fixture, handing the Cameroonian his ninth top-flight start of the season alongside Virgil van Dijk in the rearguard.
Matip's afternoon was cut short after just 69 minutes due to injury, though, and while Klopp is still unsure of the nature of his problem, he has conceded that the 32-year-old will not recover quickly.
"Of course it's much better to win a game and go into the next game. It speeds up the recovery but we lost today Joel," the Liverpool manager added.
"We have obviously no pictures yet, no scan yet but after all what you hear and can see, it will not be a short one - unlucky. But we just have to get through this.
"You have to find always a specific mood and mentality to get into the next game and to go for it and fight in hard moments and stuff like this."
Liverpool have two complete days to recuperate before they travel to Bramall Lane to take on relegation-threatened Sheffield United on Wednesday evening.