A defence-splitting pass by Mane allowed Origi to make it 1-0 on six minutes.
Xherdan Shaqiri then benefited from more classy Mane service on 17 minutes to make it two, before Michael Keane pulled a goal back for Everton four minutes later.
Origi latched on to a sublime long ball by Dejan Lovren before finishing calmly on 31 minutes.
And Mane made it 4-1 on 45 minutes, before Richarlison's strike rounded off an astonishing first half in stoppage time.
Georginio Wijnaldum slotted in from substitute Roberto Firmino's cutback late on to seal Jurgen Klopp's 100th Premier League win as manager.
Liverpool now have 43 points, while Everton drop to 18th on 14 points.
Liverpool: 7 Dec v AFC Bournemouth (A)
Everton: 7 Dec v Chelsea (H)