After failing to wrap up qualification against Toulouse on matchday four, the Reds did not enjoy a completely smooth ride against their valiant visitors, but Cody Gakpo's brace, Luis Diaz's header and Mohamed Salah's 199th Liverpool goal saw Jurgen Klopp's men get the job done with a game to spare.
With Toulouse also drawing 0-0 with Union SG in the other game in Group E, Liverpool have assured themselves of a first-placed finish in the section, thereby bypassing a playoff tie with one of the eight teams dropping down from the Champions League.
Salah and Kostas Tsimikas were the only starters from the weekend's draw with Manchester City retained for the visit of LASK, who were perhaps buoyed on by memories of taking an early lead against the Reds in September as they attacked Klopp's backline at every opportunity.
A lack of ruthlessness meant that LASK's forays forward did not bear fruit, though, while Liverpool attacked in droves themselves during an end-to-end start to the game, and the hosts needed no lessons in finding a clinical edge.
It took the group leaders just 12 minutes to open their account for the evening, as Diaz found Joe Gomez on the right and was allowed to glide into space to meet the defender's cross with a sublime header into the top corner.
Diaz was criminally allowed to ghost beyond the stagnant LASK defence unchecked, and only three minutes after the Colombian's effort, Liverpool game themselves some much-needed breathing space through Gakpo.
Diaz was involved again, trying to combine with Gakpo on the edge of the box before being crowded out, and the South American instead found the run of Salah, who delicately lifted the ball towards an unmarked Gakpo to tap in at the back stick.
The Dutchman's effort quickly sapped the life out of Thomas Sageder's side, who saw both Tsimikas and Gomez try their luck before the break, but the former's venomous attempt clattered the crossbar in the 32nd minute before Gomez - still awaiting his first Liverpool goal - fired a whisker wide of the far post nine minutes later.
Klopp sent out an identical XI for the second half, where the result was put beyond any doubt in the 51st minute, as LASK goalkeeper Tobias Lawal felled Gakpo inside the box, and Salah emphatically crashed the resulting penalty home for Liverpool goal number 199.
The Egyptian was denied the chance to bring up a double-century as he was taken off not long after, but the Reds were not satisfied to declare at three, as Gakpo rattled the outside of the post with a 25-yard piledriver in the 54th minute before Lawal denied Harvey Elliott and Darwin Nunez in quick succession.
Nevertheless, with their travelling fans in wonderful voice throughout the evening, LASK were determined to make their supporters' journey worthwhile and desperately tried to wipe out Liverpool's clean sheet, but Caoimhin Kelleher produced a strong save from the lively Moses Usor in the 76th minute.
The LASK faithful did not stop singing even as Liverpool's fourth goal flew into the back of the net in added time, as Trent Alexander-Arnold blitzed forward and laid off Gakpo, who cut inside from the left before emphatically finding the side of the net.
The Reds were forced to wait a few minutes for confirmation of their first-placed finish, but Toulouse's failure to win means that there is nothing riding on December 14's trip to Union SG for Klopp's team, while LASK cannot finish in second place but are still in contention to drop down to the Europa Conference League, sitting two points behind Union SG before their home clash with Toulouse.
Liverpool now prepare to welcome Fulham to Anfield for Sunday's Premier League clash, while LASK return to Austrian Bundesliga action away to Austria Wien in three days' time.