The hosts could not have started the game any better however, and were ahead on four minutes as Brazilian midfielder Ramires headed home for his first League goal since December 2013 to hand the Blues the lead.
Yet Liverpool worked their way back into the game, and were level just before the half-time break, Philippe Coutinho restoring parity with a fine strike to make it 1-1 at the half-time interval.
The Reds continued where they left off in the second period, with Coutinho doubling his personal, and his team’s tally with a well-placed shot on 74 minutes to put the visitors in front for the first time.
Belgium striker Christian Benteke, on as a 64th-minute substitute for James Milner, made sure of the result as he slotted home on 83 minutes to secure a 3-1 victory for the Reds, and Jurgen Klopp’s first League win as Liverpool coach.
The Merseyside club jump two place to seventh in the standings, while Chelsea suffer their sixth League loss, having won just 11 points from 11 matches.
