James Milner’s late penalty helped avoid a shock result against the relegation-threatened Cottagers after Ryan Babel’s effort cancelled out Sadio Mane’s earlier strike.
The Merseyside outfit made it difficult for themselves at times at Craven Cottage but ultimately prevailed to earn the three points that sees them dethrone Manchester City at the summit, though Pep Guardiola's side do have a game in hand.
Speaking in his post-match press conference, Klopp paid tribute to the character of his side, even as he admitted that they struggled in an often fatigued performance at times.
“Football is sometimes like this,” the German stated. “I didn’t see any panic. I liked how we reacted, but we needed a penalty. That doesn’t say we are sensational.
“The boys are human beings. I was sure we would not panic and show the nerves you are all waiting for us to show.
“We are in this situation long enough to know it is difficult for other teams to beat us, and that gives us always an opportunity to win.
“In most of the draws we had, we were closer to winning the game, and that gives us confidence to go for it and do the right things.
“For five or 10 minutes before the equaliser, we had so many mistakes, which made it a wild game. we controlled the game completely, then we gave Fulham the ball and then the crowd came and it was a bit more exciting again.
“We had a very intense week going to Munich and coming back, travelling to London. Average game, sensational result.”
Liverpool now enjoy a fortnight off thanks to the international break, though a number of their stars are expected to be in action.
One prospective doubt is England man Jordan Henderson, who hobbled off with an ankle injury in the first half of the win at the Allianz Arena in midweek.
The midfielder missed the trip to Fulham but Klopp added that he may yet play for his nation in the upcoming European Championship qualifiers.
“Hendo will go to the national team,” the manager added. “I think he will have two or three days at Melwood with his rehab.
“But it looks positive, and I think he is in contact with Gareth [Southgate] that when he can train, he goes to the national team. And if he plays, then Gareth will decide.
“It was not that serious. In three or four days, he should be fine.”