But after following up their win at fading champions Chelsea in their previous away game in the Premier League with a demolition of would-be successors Manchester City, Klopp’s Liverpool might yet be top-four material.
Liverpool’s first victory at the Etihad Stadium since 2008 means the Reds are still six points adrift of Arsenal, who occupy fourth place.
But Liverpool have now been to City, Arsenal, Manchester United, Everton, Tottenham and Chelsea already this season.
If they can get it right at home and build on the belief they should have accrued from losing only at Old Trafford from the aforementioned run of fixtures, a challenge for Champions League football in the second half of the season should not be beyond them.
“I would love to win a game like this at Anfield. That’s the next challenge. We lost to Palace (in our last home game) so we have to find a solution and we will,” said Klopp, after his side became only the second away side to avoid defeat in City’s last 14 home Premier League games.
“If you are stuck in the forest and it’s dark and you are afraid and someone tells you not to be, then it doesn’t work. It’s your own mind.
“Only you can affect that. We need the confidence in ourselves but we cannot switch the lamp on and off, on and off. So Saturday was very good but we have to work.
“The money is in the Champions League but I can’t promise it now. It’s hard work. They changed the manager here. It wasn’t because of the weather and it wasn’t a problem with Brendan Rodgers, because he is a brilliant manager.
“But they had to change something. Only two weeks ago we lost to Palace. I can’t say we are good enough, but if you want to write it you can.”
Klopp has managed to change something and against a rampant Liverpool side, City couldn’t cope.
Missing Vincent Kompany and resting Nicolas Otamendi following his exertions with Argentina during the international break, City manager Manuel Pellegrini selected a back four that can never have endured such a collective break-down as Liverpool went 3-0 up after just more than half an hour.
Bar a slight wobble following Sergio Aguero’s excellent response, they could easily have won by more.
Pellegrini refused to blame only his back four on a dark night for a team that never looked likely to regain the top spot they had relinquished earlier in the day to Manchester United and then Leicester City.
“It’s really unbelievable the way we played,” admitted Pellegrini, who conceded the performance had been the worst of his two-and-a-half year reign.
“Sometimes you can be bad defending and concede easy goals and you lose the game. But to make it all wrong in attacking, defending and possession, it’s not the team we know.”
City were dreadful but might have got away with it against a side not as ruthless as Liverpool. Klopp’s mission for high-tempo, pressing football could not have been better illustrated with Emre Can and Lucas Leiva superb in midfield behind a quartet of James Milner, Adam Lallana, Roberto Firmino and Philippe Coutinho, who never gave City a moment’s piece.
Coutinho and Firmino combined for the opening goal after seven minutes when Coutinho robbed Bacary Sagna and released Firmino to deliver a low cross which was miscued into his own net by Eliaquim Mangala.
Neither of City’s French duo recovered and Sagna admitted afterwards that the Paris terror attacks had taken their toll. “Personally it’s not my best week. But it’s life, and there’s nothing you can do about it. It’s part of the past now and just have to keep your head up. It was not easy to be part of that event.
“Mentally I was (ready to play). Physically I believe I was. But I wasn’t. I tired quite quickly and didn’t have as much power as I used to.”
Mangala was also partly at fault for Liverpool’s second goal, finished off by Coutinho to give the Brazilian a fourth goal in three games. Firmino then got his first following outstanding play from Can and Coutinho.
Firmino could have had a hat-trick but for the belated excellence of Joe Hart but further goals weren’t needed as City failed to build on an excellent strike from Aguero just before half-time.
Raheem Sterling had a game to forget against his former club as he passed up City’s one real chance of a comeback early in the second half and Liverpool’s superiority was confirmed late on by Martin Skrtel, whose thumping volley epitomised his command performance.
Hart 7; Sagna 4, Demichelis 5, Mangala 4, Kolarov 6; Toure 5 (Fernandinho ht, 6), Fernando 5; Navas 4 (Delph ht, 6), De Bruyne 6, Sterling 5; Aguero 6 (Iheanacho 65, 4).
Mignolet 7; Clyne 8, Skrtel 9, Lovren 8, Moreno 8; Leiva 8, Can 8; Lallana 8 (Toure 90), Milner 8, Coutinho 9 (Ibe 68, 7); Firmino 8 (Benteke 76).
Jonathan Moss 8.