However, despite Michael Keane and Richarlison contributing for Everton before the break, the Toffees struggled to trouble Liverpool for large periods of the second half, with the hosts completing the triumph through a late Georginio Wijnaldum effort.
While Liverpool stay in control in first place, as well as extending their unbeaten home league record over their rivals to 20 games, Everton drop into the relegation zone, inevitably leaving Marco Silva on the brink of losing his job in the dugout.
The first goal of a remarkable opening 45 minutes went the way of the home side, as Origi rounded Jordan Pickford to slot into an empty net after latching onto a sublime through-ball from Mane.
While Everton had not gone into their shell, it had quickly become obvious that Liverpool were relishing the opportunity to find space in behind the visitors' backline, which soon led to the top-flight frontrunners extending their advantage.
After receiving a pass from Trent Alexander-Arnold, Mane bided his time before picking out the run of Shaqiri, who was able to direct a low, bobbling effort into the far corner from eight yards.
However, while the hosts had ambitions to further extend their lead, Everton quickly halved the deficit with Keane producing a deft finish over Adrian after the ball had fallen for him inside the penalty area.
Late challenges from Alexander-Arnold and Richarlison resulted in yellow cards as tensions threatened to boil over, although it was not long before Liverpool reopened their two-goal lead.
A precise long pass from Dejan Lovren picked out the run of Origi, who brought the ball down with one touch before firing a shot over Pickford with his second from inside the area.
With eight minutes still remaining of the first half, Liverpool further extended their advantage through Mane, who ended a lightning counter-attack involving Alexander-Arnold by sweeping a left-footed shot into the bottom corner from 20 yards.
While it was a bold move by the under-pressure Portuguese, it eventually paid off in added-on time as the substitute provided a cross for Richarlison to send a header into the far corner of Adrian's net.
Despite many expecting more of the same after the restart, clear-cut opportunities were few and far between with only a Lucas Digne free kick threatening a seventh goal of an absorbing contest.
Nevertheless, Liverpool were comfortable with seeing plenty of the ball in the Everton half of the pitch, earning free kicks at regular intervals to help wind down the clock.
Just after the hour mark, the Toffees created another opportunity from a wide area, with Keane sending a header into the grateful hands of Adrian after meeting an outswinging corner from the right.
That approach almost worked with Mane being presented with an opportunity after controlling a long ball, only for the Senegalese to somehow fire wide with just Pickford to beat.
Moise Kean should have at least tested Adrian when getting in behind the Liverpool backline soon afterwards, instead shooting wide from the edge of the area, and the home team eventually capitalised in the final minute of normal time.
After fine work from Firmino on the left-hand side of the penalty area, Wijnaldum was able to take a touch before driving the ball into the far corner of the net from 15 yards out.
As Liverpool and their supporters celebrated a 14th win from 15 outings in the Premier League this season, the majority of the travelling fans had already left by the full-time whistle after witnessing their team drop down into 18th position in the table.
LIVERPOOL (4-3-3): Adrian; Alexander-Arnold (Gomez 83'), Lovren, Van Dijk, Robertson; Wijnaldum, Lallana (Henderson 72'), Milner; Shaqiri, Mane, Origi (Firmino 73')
EVERTON XI: Pickford; Sidibe (Bernard 35'), Holgate, Mina, Keane, Digne; Iwobi, Davies (Schneiderlin 72'), Sigurdsson, Richarlison; Calvert-Lewin (Kean 60')