Brighton and Hove Albion 1 Liverpool 5: Firmino at the double for Klopp's counter-attacking Reds
The Reds looked to be out of sight when they netted twice in two first-half minutes and then extended their lead further after the break, but Glenn Murray pulled a goal back to threaten a fightback.
It never materialised, though, as Jurgen Klopp's men turned on the style in the closing stages.
Brighton have made the Amex Stadium somewhat of a fortress this season, previously losing only to Manchester City, but they could not deny Liverpool a victory that was set in motion by Emre Can's 30th-minute opener.
Firmino's first swiftly followed as Brighton struggled to cope with the speed of the visiting attack, with Klopp's side then adding a third - through the Brazilian forward again - after the restart.
Again there was a second strike in quick succession, but this time it was Brighton's - Murray netting from the penalty spot having twice earlier squandered big chances.
Murray could have scored again, too, but Liverpool added an extra gloss to proceedings with two late goals, both inspired by Philippe Coutinho, as the Reds moved above Arsenal and into the top four.
Boom! pic.twitter.com/OH6KzF4azU
— Liverpool FC (@LFC) December 2, 2017
Coutinho and Mohamed Salah were among six changes made by visiting boss Klopp from the midweek defeat of Stoke City, including Can and Georginio Wijnaldum in a makeshift three-man defence - largely due to Joel Matip's absence.
And Salah was quickly in the thick of an action, dashing down the left to force a corner from which Firmino wastefully nodded over the top.
Brighton's determined early stance contributed to limited attacking opportunities at either end, but Murray might have done better when he turned and blasted off target from inside the area in a rare opening for the home side.
Can broke the hosts' resolve on the half-hour mark, though, climbing to head home a set-piece from Coutinho, who quickly supplied the second, too.
The hosts were still reeling from the opener as Salah and Coutinho broke on the left and Firmino arrived in the box to meet the latter's centre for a simple - yet devastating - finish.
The game then seesawed dramatically as the second half set off at a frantic pace, with Simon Mignolet brilliantly denying Murray from close range before Firmino's second followed at the other end.
It was another blistering Liverpool counter and the slaloming Salah teed up Firmino for a simple finish, but the three-goal margin lasted only three minutes.
15 - Mohamed Salah has had a hand in 15 goals in his first 15 Premier League games for @LFC - more than any other player for the club. (12 goals, 3 assists). Talisman. pic.twitter.com/6iYU71oHsd
— OptaJoe (@OptaJoe) December 2, 2017
Referee Graham Scott awarded what appeared to be a soft penalty against Reds captain Jordan Henderson as he tussled with Shane Duffy, allowing Murray to finally find the net from 12 yards.
Mat Ryan smartly prevented Salah from adding his name to the scoresheet with a left-footed effort following Firmino's pass and the goalkeeper was soon tested again by the influental Coutinho.
Murray's snapshot on the turn fizzed wide as Brighton kept pushing, but another flurry late on put a third consecutive away league win beyond doubt for Klopp's men.
Coutinho's magnificent free-kick crept under the wall for the fourth and the Brazilian magician forced the fifth, too, as Lewis Dunk deflected a shot beyond Ryan in the dying seconds.