Lewis Koumas has been an energetic presence for Liverpool in the absence of Alexander Isak, and he made a nuisance of himself early on, running onto a Jeremie Frimpong through ball.
He rounded the goalkeeper, but the angle was always closing, and the Austrian defence was able to see it out for a corner.
Florian Wirtz then went close, dancing through defenders in the box and almost squeezing his finish under the goalkeeper.
The breakthrough looked inevitable, and it came through Ronald Araujo, who rose highest from a corner to head home his first goal for the club since arriving on loan from Barcelona.
One became two in the blink of an eye. Rio Ngumoha ducked and weaved through the home side, and might have had a penalty shout.
But the ball broke to Koumas, who shuttled it towards Barcola, with the Frenchman sweeping it home to open his Liverpool account as well. Koumas briefly thought he had got in on the action himself, swivelling sharply and slotting the ball into the corner, but he was narrowly offside.
Ryan Gravenberch then struck the inside of the post as the hosts stayed in the contest by the skin of their teeth.
Christoph Lang dragged an effort wide in a rare foray forward, but soon it was 3-0, as Barcola latched onto an excellent Wirtz pass and added his second of the game.
LASK looked a little livelier after the interval, and Lang continued to pose the biggest threat, flashing an effort narrowly wide of Alisson Becker’s post.
But any distant hope was short-lived, as Frimpong scored the first headed goal of his career to extend the visitors’ advantage still further.
The unfortunate Koumas was then denied by the offside flag for a second time, before substitute Robert Ljubicic pulled one back for the Austrians, controlling the ball with his first touch of the match and smashing it in with his second.
But it was Liverpool’s night, and Barcola’s especially, with the 24-year-old sweeping in Frimpong’s cross to complete his first professional hat-trick.
Koumas then finally got in on the action, collecting yet another Frimpong delivery and scuffing it through the goalkeeper’s legs to make it 6-1.
There was another twist to come, however, with substitute Florian Flecker scoring a brace to peg it back to 6-3, going one better than his goal against the Merseysiders in 2023.
He was on hand to head into an empty net at the far post with his first, before controlling a nice strike into the corner to irk Andoni Iraola and pose some kind of question in the closing stages.
But the Reds fended off any real threat of a comeback to move to 26 wins in their last 30 Champions League group/league phase matches, while the wait for an Austrian side to beat an English opponent in a major European competition looks set to extend beyond 35 years.
Flashscore Man of the Match: Bradley Barcola (Liverpool)