Klopp would have been delighted by the way his side began the latest match in his swansong tour, with the movement of Luis Diaz and Darwin Nunez, in particular, causing the Chelsea defence all manner of problems.
The latter was picked out by Conor Bradley, but his looping effort was kept out by a scampering Dorde Petrovic.
Núñez was denied twice more by a combination of the woodwork and the Serbian shot-stopper in the opening exchanges before the inevitable Liverpool breakthrough with 23 minutes on the clock.
Diogo Jota showed brilliant persistence to force his way between Thiago Silva and Benoit Badiashile, firing home past Petrovic to open the scoring.
The odds were firmly stacked against a Chelsea victory now, given that Liverpool had won 77 and drawn nine of their previous 86 PL matches at Anfield when scoring first.
There was a landmark moment in the 39th minute when Bradley doubled the Reds’ advantage with his first senior goal for the club, following a brilliant angled finish from a Díaz pass.
The hosts squandered a chance to make it 3-0 before HT after Badiashile was penalised for standing on Jota’s foot in the box, as Nunez hit the woodwork again from the resulting spot-kick.
Malo Gusto and Mykhailo Mudryk were both introduced after the break to add fresh impetus to the Blues’ impotent attack, and the duo should have combined to halve the deficit. The Frenchman picked out Mudryk inside the box, but the Ukrainian blasted his effort high and wide of the target.
But just when the Blues were showing flickering signs of a revival, Liverpool added a third through Dominik Szoboszlai, who headed home Bradley’s superb cross from the right-hand side.
The deficit was cut back to two though courtesy of Christopher Nkunku’s smart finish following a fine run by Carney Chukwuemeka. Mauricio Pochettino was incandescent with rage on the touchline when Chelsea was denied a penalty after Virgil van Dijk appeared to bring down Nkunku in the box.
Nevertheless, the game continued, and Díaz appeared to end any faint hopes of a Chelsea comeback when the Colombian slid the ball home from a Núñez cross.
This Liverpool victory ended a run of seven consecutive draws between the two sides across all competitions - the longest streak of stalemates between top-flight clubs in English football history.
Meanwhile, there was more misery for Pochettino against Klopp, having now won only one of his 13 league and cup battles with the German.
Flashscore Man of the Match: Conor Bradley (Liverpool)
