Liverpool continued their phenomenal goalscoring away from home by hitting five at Brighton to move into the top four of the Premier League for the first time since August.
Brighton had chances to make it more nervy for Liverpool with a second goal, but Coutinho eased any concerns about a repeat of Sevilla's comeback from 3-0 down in the Champions League by scoring with a cute low free-kick then enticing Brighton defender Lewis Dunk to head into his own net.
Klopp's side move above Arsenal, who host Manchester United at 5.30pm.
Jurgen Klopp's adventurous side have scored 15 goals in their past four away games in all competitions and continued their resurgent league form with a comfortable victory at the Amex Stadium.
Emre Can powered a header beyond Mat Ryan from Philippe Coutinho's corner after half an hour and the Reds doubled their lead 79 seconds later following a sumptuous move.
Mohamed Salah drove at the Brighton defence and picked out Coutinho, who slid a left-footed cross to the back post for Roberto Firmino to finish efficiently.
Brazil striker Firmino made it 3-0 after Salah's brilliant assist before Glenn Murray, who had already missed two great chances, beat Simon Mignolet with a penalty following Jordan Henderson's foul on Shane Duffy.
This was another masterclass in the art of scoring by the Reds, but it was also a triumph of sorts at the other end of the pitch.
Klopp was without several defenders - Jose Gomez (virus) and Joel Matip (injured) were unavailable, Alberto Moreno was rested and Ragnar Klavan (ill) only on the bench - meaning the German fielded an unfamiliar back line in terms of both formation and personnel.
Liverpool lined up with three centre-backs - and one of those was Georginio Wijnaldum, whose reputation was formed as a No.10 in the Netherlands and who had been brought to Anfield in July 2016 to score goals from midfield.
Klopp also selected midfielder Can as the right-sided centre-back and handed left-back Andrew Robertson his first Premier League start since 16 September.
So it was perhaps inevitable Liverpool wouldn't be rock solid at the back and fortunately for them, other than his penalty kick, Murray was profligate when the re-jigged defence permitted the home side chances.
Brighton frustrated Manchester United for more than an hour at Old Trafford in their previous game before Dunk's unfortunate own goal proved the difference between the sides.
This time, the Seagulls were well on their way to defeat when the defender's header made it 5-1 in the 89th minute.
Dunk also put past his own keeper on the opening weekend of the season against Manchester City, and is now one own goal away from equalling ex-Reds defender Martin Skrtel's record of four in a single Premier League season - which the Slovakia centre-half did in 2013-14.
Brighton boss Chris Hughton described Liverpool as the best offensive side his side had faced since Pep Guardiola's City after the game.
But his team helped their opponents to victory. In addition to Dunk's gift, the wall for Coutinho's goal failed miserably to do its job, Can out-jumped a static home defence for his first league goal of the season and they were hopelessly caught on the counter attack for Firmino's second goal.
Brighton had not conceded more than two goals in a game before this match and their response to this emphatic defeat will be a first big test of character since winning promotion to the Premier League.
Klopp made a further six changes to his side following the 3-0 win at Stoke City on Wednesday.
Liverpool have made a total of 54 changes to their starting line-up this season in 15 Premier League games - more than any other side.
Their fifth win in six top-flight games was achieved without Senegal winger Sadio Mane or England striker Daniel Sturridge making it onto the pitch from the bench and England midfielder Adam Lallana not available after his long lay-off.
Klopp is likely to rotate his squad again for the Champions League game with Spartak Moscow next Wednesday as well as in the Merseyside derby at Anfield on 10 December.
At the moment, it doesn't seem to matter which of his attacking players he fields away from home - lots of goals and positive results seem to follow.
Man of the match - Philippe Coutinho (Liverpool)
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