Diogo Jota opened the scoring with a solo effort before Conor Bradley scored his first senior goal for Liverpool to leave Chelsea behind at the break for the first time in the Premier League this season.
Darwin Nunez hit the post with a penalty before the half was over, one of four occasions on which he struck the woodwork - the most on record in a Premier League game.
Christopher Nkunku pulled one back for Mauricio Pochettino's side and was very unlucky not to be awarded a penalty soon after, but Luis Diaz ended any hope of a comeback with the fourth.
Jurgen Klopp did not have to wait long to see his side make it three in his first Premier League game since announcing his decision to leave at the end of the season
The result sees Chelsea stay 10th in the table, their best hope of a successful season likely to come in the Carabao Cup final next month.
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