The Reds’ front three of Mohamed Salah (two), Roberto Firmino and Philippe Coutinho got the goals as they put their opponents to the sword in the first half with Firmino adding a fifth shortly after the break and deadline-day signing Alex Oxlade-Chamberlain scoring his first for the club late on followed by Trent Alexander-Arnold’s deflected added-time effort.
Their previous best win away from home in Europe had been 5-0 - against Reykjavik in 1964, Crusaders in 1976 and Haka in 2001 - but the Slovenians made that record easy to beat.
It was Liverpool’s first away win in the Champions League since the 1-0 victory against Debrecen in November 2009 when David Ngog scored the only goal.
On that occasion it could not prevent their exit from the competition.
By contrast this result may be the one which gives them the push towards the knockout stage as, after two disappointing draws, they now top Group E on goal difference from Spartak Moscow with a return game against Maribor in a fortnight a chance to strengthen their position.
In a start as bright as their ‘bold citrus’ (orange) third kit the visitors were ahead in four minutes when Salah latched onto Mitja Viler’s mistake to outsprint Marko Suler to the ball close to the right touchline and then cross for Firmino to sidefoot home.
James Milner, impressing in only his third start of the season in place of captain Jordan Henderson, crossed for Coutinho to cushion a 13th-minute volley past Jasmin Handanovic for his fourth consecutive away goal, a feat last registered by Steven Gerrard in March 2014.
Firmino’s clever through-ball inside left-sided centre-back Suler allowed Salah to open his body and tuck a left-footed shot inside the far post for his seventh of the season as Liverpool scored three from their first four shots on target, reversing a worrying trend which developed in September.
Coutinho and Alberto Moreno combined for Salah to nip in front of Firmino to bundle home his second - the first time since December 2006 they had scored four first-half goals away from home and the first English club to do it in an away Champions League tie.
It could easily have been more as Coutinho and Salah both had chances saved by Handanovic, although Suler should have reduced the deficit but planted a free header wide.
There was no respite with Firmino heading in Coutinho’s free-kick to make it 5-0 early in the second half, from which point Klopp was able to rest Salah, Firmino and Georginio Wijnaldum and give meaningful run-outs to Alex Oxlade-Chamberlain, Daniel Sturridge and Dominic Solanke.
The former two combined for the ex-Arsenal to grab his maiden Liverpool goal before Alexander-Arnold drove home via a deflection.
Handanovic, Milec, Rajcevic, Suler, Viler, Kramaric (Hotic 58), Kabha, Vrhovec, Bohar (Mesanovic 81), Ahmedi (Pihler 58), Tavares.
Karius, Alexander-Arnold, Matip, Lovren, Moreno, Can, Wijnaldum (Solanke 76), Milner, Salah (Oxlade-Chamberlain 57), Firmino (Sturridge 68), Coutinho.
Viktor Kassai (Hungary).