Liverpool broke the deadlock in the 14th minute through Trent Alexander-Arnold, who curled his effort into the top corner after Mohamed Salah rolled the ball to him at a free-kick.
In the 30th minute Liverpool doubled their lead after Roberto Firmino rose highest to head in Andy Robertson’s cross.
Chelsea came into the second half more purposeful and were rewarded in the 71st minute off N’golo Kante who surged into the area and unleashed an unstoppable shot into the top corner.
Chelsea continued to press and could have levelled up on 88 minutes only for Michy Bashuayi’s header from range just missing the target.
The Reds remain top on 18 points while Chelsea are 11th on eight points.
They are now the first team in top-flight history to win their opening six matches of a top-flight season in two consecutive seasons.
And at the Emirates, 10-man Arsenal beat Aston Villa 3-2 to move into the top four.
Arsenal found themselves behind on 20 minutes as John McGinn timed his run to and diverted Anwar El Ghazi’s cross into the net.
With one minute left in the first half, Arsenal were reduced to 10 men following Aisley Maitland-Niles’s sending off.
In the 59th minute Arsenal equalised thanks to Nicolas Pepe from the penalty spot after Mateo Guendouzi was fouled inside the box.
Just within seconds of conceding an equaliser, Wesley was on hand to deftly flick home Jack Grealish’s cross in the 60th minute to put Villa 2-1 ahead.
With nine minutes remaining in the game, Arsenal were back levelled through Calum Chambers who reacted quickest to a loose ball and flicked it into the roof of the net.
And in the 84th minute, Arsenal got the winner off a Pierre-EmerickAubameyang’s superb free-kick.
Arsenal are now on 11 points while Villa are 18th on fourpoints.