Klopp's side made it six successive league wins to open the season as they survived a second-half Chelsea barrage to move five points clear of Manchester City.
Trent Alexander-Arnold's blistering free-kick and a Roberto Firmino header put Liverpool in control before N'Golo Kante's superb second-half strike set up a tense finish.
Liverpool managed the pressure-packed situation with the kind of mature, well-drilled display that is the hallmark of champions.
And Reds boss Klopp insisted he never doubted his players would thrive under the cosh because they are such a committed group.
"I shouldn't be surprised anymore by the character of our team. We expect it from us that we fight for everything," he said.
"That is how it is in the top six and in the Premier League. You will never win at Chelsea without putting a proper shift in and we did that today."
After City's astonishing 8-0 demolition of Watford on Saturday, this was the perfect response from Liverpool as they try to dethrone Pep Guardiola's champions.
Chasing a first English title since 1990, Liverpool are on a club-record run of 15 successive league wins and have lost just once in 45 top-flight matches, going unbeaten in the last 23.
"We knew how good City were before the game yesterday. I really believe they are the best team in the world.