Jurgen Klopp's side were only slightly deserving of the three points as the Champions League champions defeated Chelsea 2-1 on Sunday afternoon at the Stamford Bridge.
Both sides came off the back of stinging losses in the Champions League in midweek but it was the Reds who recovered better to post a narrow victory at the home of the current holders of the Europa League trophy.
Chelsea had more possession, earned more corner kicks, fired more shots and made more passes, but two first half goals from Alexander-Arnold and Firmino got Liverpool back to winning ways and send them five points clear of Manchester City, after the champions walloped Watford 8-0 on Saturday evening.
The match started on an even note but it was Liverpool who, having taken their chances in the first half, lead Chelsea 2-0 at the break.
English man Trent Alexander-Arnold opened the scoring with a brilliant goal after Salah laid off a free-kick to him and he fired it into the top right corner.
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Chelsea thought they had equalised through Azpilicueta just before the half-hour mark, but VAR overruled the decision as Mount had been offside in the build-up.
Brazil international Roberto Firmino then doubled Liverpool's lead minutes later when he headed Robertson's cross past Kepa and into the back of the net after just half an hour gone.
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Chelsea showed more urgency after the break with Abraham and Mason Mount going close within minutes of each other but the hosts could have gone further behind. Goalkeeper Kepa was to thank, though, for twice coming up with crucial saves to keep the score line respectable.
With twenty minutes of regulation time left, midfielder Ngolo Kante scored a beauty to drag the Blues back in the game. The French man turned past Fabinho wonderfully and then was not closed down as he made the run to the edge of the box and set himself for the shot, a sublime effort that flew past Adrian and into the top right corner.
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Chelsea poured forward in search of an equaliser and with the last throw of the dice, Frank Lampard withdrew the ineffective Tammy Abraham and sent in Belgian Batshuayi.
But it was too little too late as Jurgen Klopp's men had done just enough in the first half to bag all three points at stake.
