The Reds were three goals to the good within 20 minutes en route to their first Champions League victory of the season, with the front three of Roberto Firmino, Philippe Coutinho and Mohamed Salah all on the scoresheet for the rampant visitors.
Liverpool's Champions League campaign before tonight had been a tale of frustration and missed chances, but that was not the case this evening and it took less than four minutes for the visitors to break the deadlock when Salah tore down the right flank before laying it on a plate for Firmino to end his five-match wait for a goal.
James Milner - making just his third start of the season in place of captain Jordan Henderson - was the next to come close as Liverpool pushed for a quickfire second, but his low strike was kept out by Jasmin Handanovic.
Even at that early stage of the match it looked as though it could be a long night for the Maribor keeper, and he was left to pick the ball out of the back of his net again after 13 minutes as Coutinho latched on to Milner's cross to fire a bobbling effort into the bottom corner.
The Reds were rampant and should have had a third just two minutes later when Firmino flicked an effort narrowly wide of the far post, before Alberto Moreno slammed a powerful strike of his own into the side-netting shortly afterwards.
A third goal was inevitable, and Liverpool killed off the match as a contest within 20 minutes when Salah got in on the act with a cool finish past Handanovic having been played through by Firmino.
The visitors' front three were all already on the scoresheet and continued to cut Maribor open almost at will as Salah drew another save after being found by Coutinho's pass.
Maribor finally tested Loris Karius for the first time shortly before the half-hour mark when the Liverpool keeper got down well to deny Mitja Viler, but the visitors were quickly back on the front foot and Salah was again kept out by Handanovic moments later.
Coutinho was the next to threaten after creating space to shoot with a mazy run inside the box, and a fourth did arrive for the Reds six minutes before half time when Moreno's low cross was tucked in by a mixture of Firmino and Salah - the latter seemingly getting the final touch from close range.
The hosts should have had a consolation shortly afterwards as the action continued to arrive thick and fast, but Marko Suler planted his header wide of the target having been left unmarked to latch on to Valon Ahmedi's free kick from deep.
Any forays forward for Maribor were fleeting, though, and there was still time for two more Liverpool chances before half time as first Firmino and then Coutinho came close to adding a fifth goal in the opening 45 minutes.
Klopp's side quickly resumed their dominance upon the restart, with Moreno drilling one low strike wide of the target before Georginio Wijnaldum and Coutinho were then denied within one minute of each other.
The fifth goal did arrive nine minutes into the second half when Firmino doubled his own personal tally for the evening, glancing a deft header into the far corner from Coutinho's free kick.
The five-goal margin drew Liverpool level with their biggest ever European away win, and it was also the signal for Klopp to look ahead to Sunday's Premier League clash with Tottenham Hotspur as both Salah and Firmino were brought off while on a hat-trick.
Oxlade-Chamberlain replaced Salah and almost made an impact within 30 seconds of his introduction when he latched on to Moreno's delivery, but he could not steer an ambitious first-time volley on target.
Oxlade-Chamberlain came close again with a little under 15 minutes remaining, collecting a poor clearance from Handanovic before bursting past his marker and drilling a low shot towards goal which the Maribor keeper had to deal with.
The former Arsenal midfielder did finally open his Liverpool account with four minutes of normal time remaining, though, as another defensive error saw Daniel Sturridge lay the ball into his path, and this time he made no mistake with his finish.
Liverpool were not done there, though, and a seventh goal capped off the scoring in the 90th minute as Alexander-Arnold got in on the act when his strike deflected off the hapless Suler and past Handanovic.
The Reds fell just one goal short of equalling their own all-time Champions League record for the biggest win in the competition, but the seven-goal winning margin means that it is the biggest away win from an English team in European Cup or Champions League history.
Results elsewhere also went in Liverpool's favour on the evening, with Spartak Moscow's surprise 5-1 thrashing of Sevilla seeing the Reds leapfrog the Spanish outfit into top spot of Group E, with just one point now separating three teams.