Antonio Conte suffered his first defeat as Chelsea manager as Liverpool moved alongside their hosts in the top four of the Premier League.
Defender Dejan Lovren was left unmarked to volley in Philippe Coutinho's left-sided cross for the Reds' opener.
Skipper Jordan Henderson doubled the visitors lead with a stunning strikeย from 30 yards out, leaving keeper Thibaut Courtois completely helpless as the ball sailed into the top right corner.
Chelsea improved after half-time as Diego Costaย tucked in Nemanja Matic's cutback to score his fifthย goal of theย season, but Liverpool held on to join theย Blues - and Everton - on 10 points.
Jurgen Klopp's side have already also beaten Arsenal and Premier League Champions Leicesterย City, plus takenย a point at Tottenham, with their only defeat coming at newly-promoted Burnley.
Both sides now turn their attention to the League Cup in midweek with Chelsea travelling to play Leicester and Liverpool on the road against Championship side Derby County.
Next up in the leagueย forย Chelsea next week is the London derby with Arsenal, while Liverpool host Hull City.
Full Time
Chelsea (Costa 61') 1 Liverpool 2 (Lovren 17' & Henderson 36')
Tomorrow Morning
Hull City vs Arsenal - 12:00am AEST
Leicester City vs Burnley - 12:00am AEST
Manchester City vs Bournemouth - 12:00am AEST
West Bromwich Albion vs West Ham United - 12:00am AEST
Everton vs Middlesborough - 2:30am AEST
