Liverpool got their third after the break as Jordan Henderson swept in an excellent cross for Sadio Mane, who squeezed his header past Claudio Bravo in goal.
Klopp praised his side for scoring when it mattered, insisting they put in the hard graft to deserve their victory.
“I like the game. I love the atmosphere, it was incredible, people just amazing, against an outstandingly strong opponent,” Klopp told reporters.
“The second goal we scored, I don't think I ever saw a goal like this. Probably not.
“A right full-back with a 60-yard pass to the left full-back; two more touches and a cross over 40 yards, 50, and a header. That's pretty special.
“It was a good moment to score a goal like this. And the third goal very, very good as well, a sensational cross and Sadio in the right position.
“That was all good, and all-around these three goals it was the hardest work, and so the boys deserved these three points. Great.”
The manager also had warm words for Sean Cox, who has spent the last 18 months receiving treatment since sustaining life-changing injuries after being attacked by Roma fans in April last year.
“The love that Sean and his family has for this club, and for football, I hope that it gave them all strength and hope in the recovery time,” Klopp added.
“Having this can give him the opportunity, them the opportunity, to come back and watch such a really big game.
“That he wants to do that, for me, coming from a really low point, is one of my highlights since I'm here.”
Meanwhile, Guardiola was peeved with the referee and VAR's decision to not award a handball against Trent Alexander-Arnold, moments before Fabinho's opening strike for the Reds.
“Ask the referees, don't ask me, he said,” when questioned about the decision.
“Ask Mike Riley and the guys in VAR. Don't ask me, we tried to do our job.
“I know the way teams play when they come here and have so much respect for Anfield, and with the problems we had in the squad, the way we played was awesome.
“It was one of the best performances we have played. We played like back-to-back champions. That is the point. And if in seven months time Liverpool win, I will be the first to congratulate them.
“We cannot deny how good Liverpool are but the way we played, the way we created chances, the personality, it was good.
“At the end, in the boxes, it was quite similar to the Champions League, three shots on target, and the first two were goals, so it's not easy. But after that, we continued and we did it.”
The Reds face Crystal Palace at Selhurst Park after the international break, with home fixtures against Napoli in the Champions League and Brighton in the Premier League to round off an another excellent month, thus far.