Six goals were scored in a remarkable first half, with Liverpool - who started with Mohamed Salah and Roberto Firmino on the bench - supremely clinical in attack, Origi striking either side of Xherdan Shaqiri's first goal of the season.
Origi's second came after Michael Keane had pulled one back for Everton, but Mane - provider for Liverpool's first two goals - made it 4-1. Richarlison reduced the deficit again in first-half stoppage time, but it was not enough to inspire an Everton comeback after the break.
Instead, Georginio Wijnaldum's late strike rounded out the scoring, leaving Marco Silva's future looks increasingly uncertain as Klopp's side extended their unbeaten league run to 32 matches, a new club record.
Origi, Liverpool's hero in the corresponding fixture last term, wasted little time in getting things started, coolly rounding Jordan Pickford after latching onto Mane's exquisite pass.
Mane was the recipient of an even better pass for Liverpool's second goal - Trent Alexander-Arnold picking out the winger with a sensational diagonal - that saw Shaqiri slide a low effort across Pickford and into the net.
Keane's neat finish over Adrian hauled Everton back into the contest in the 21st minute, only for Origi to restore Liverpool's two-goal cushion.
Divock. Origi. 👏 pic.twitter.com/ltWk97jtGb— NBC Sports Soccer (@NBCSportsSoccer) December 4, 2019
Liverpool rounded off their rampaging first-half display when Mane slotted in from the edge of the area at the culmination of another sweeping counter, yet Everton had a second when Richarlison diverted Bernard's cross home via his shoulder.
However, the second half did not match the frantic pace of the first. Substitute Moise Kean had a glorious chance to make Mane pay for missing two golden opportunities, but drilled wide when one-on-one with Adrian.
Wijnaldum showed no such profligacy, however, dealing the final blow to Silva's team as Liverpool remain eight points clear at the top thanks to a victory that sends Everton into the relegation zone.
Wijnaldum gets his goal 🙌 pic.twitter.com/8d4HWKr1ru— NBC Sports Soccer (@NBCSportsSoccer) December 4, 2019
What does it mean? Record-setting Klopp marks a stark contrast to despairing Silva
Just over a year ago, Everton - who have not won at the home of their neighbours in the top flight since September 1999 - were Liverpool's equals for much of the Anfield derby, only to be outdone by Pickford's calamitous error.
But now, with Everton in the bottom three, Silva looks all set to be heading for the exit door. Opposite number Klopp, meanwhile, is the second quickest - after Jose Mourinho - to reach 100 Premier League victories.
Origi's derby run rolls on
Liverpool's Champions League hero seems content with a back-up role, yet he seems to be the man to call on when it comes to derbies. The Belgium forward has managed five goals against Everton, leaving him behind only club legends Robbie Fowler and Steven Gerrard in the Premier League era.
Sidibe the fall guy for Silva's desperate roll of the dice
Djibril Sidibe had a torrid 35 minutes defensively, with Mane capitalising on the acres of space afforded to him by Everton's wing-back. In response, Silva elected to switch systems completely, replacing the France international with Bernard 10 minutes before the break, though it did little to stem the tide.
Liverpool travel to the south coast to face struggling Bournemouth on Saturday, with Everton hosting Chelsea at Goodison Park.