First-half goals from Trent Alexander-Arnold and Roberto Firmino put the Reds two goals to the good at the interval on Sunday.
Chelsea - earlier denied an equaliser by VAR - fought back strongly in the second period, however, and halved the deficit with a little under 20 minutes left to play through N'Golo Kante.
Klopp's team then successfully withstood late pressure from the hosts to claim a sixth successive Premier League win at the start of 2019-20 and 15th overall in the top flight - a result that restores their five-point cushion at the top of the table.