Andoni Iraola's side took a first-half lead in Austria, through Ronald Araujo's first goal for the club, when he headed in from Kostas Tsimikas' corner.
BBC Sport - Bradley Barcola scored a fine hat-trick as Liverpool eased past LASK in a Champions League game that had nine goals.
Andoni Iraola's side took a first-half lead in Austria, through Ronald Araujo's first goal for the club, when he headed in from Kostas Tsimikas' corner.
Two minutes later, Barcola scored his first Liverpool goal since joining from PSG for £123m in the summer.
Lewis Koumas teed up Barcola after Rio Ngumoha made a brilliant run down the right and the Frenchman finished wonderfully with his right-foot.
Barcola - who failed to score or assist in any of his first six games - scored another just before half-time after he was played through by Florian Wirtz, before Jeremie Frimpong made it four with a header from a Tsimikas cross after the break.
LASK pulled one back through Robert Ljubicic but three minutes later, Barcola completed his hat-trick when he got on the end of Frimpong's ball, much to the delight of Liverpool's travelling support.
Lewis Koumas then got in on the act as Frimpong got another assist but the home side restored some pride through two goals in quick succession from Florian Flecker to make it 6-3 on the night.
Iraola - who made four changes from Sunday's defeat at Manchester City, including resting Virgil van Dijk - has now seen his side win two out of two in the Champions League.
Liverpool travel to Brentford in the Premier League on Saturday before taking on Villarreal at Anfield next Tuesday.