Liverpool delivered a much improved performance in the second half to take all three points in Austria
LASK midfielder Florian Flecker struck a brilliant opener from the edge of the area on 19 minutes for the Austrians.
Darwin Nunez equalised from the penalty spot after Luis Diaz was fouled, before Diaz himself scored the winner from debutant Ryan Gravenberch's cross.
Second half substitute Mohamed Salah added a third goal late on for Jurgen Klopp's side to seal victory.
Klopp made 11 changes to the Liverpool team which started against Wolves in their previous fixture.
The impact of those changes was clearly felt as the Reds endured stumbling first-half display in their first Europa League match since the 2016 final defeat to Sevilla, before an improved performance after the break helped them turn the match around.
It meant Klopp became the first Liverpool manager to win 50 European games.
But he said of the landmark: "Let me say it like this. If I still have 50 after the group stage, then even if I've still got the most wins in Europe as a Liverpool manager then everybody will hate that."
Elsewhere in Group E, Union St-Gilloise and Toulouse drew 1-1, leaving Liverpool two points clear at the top of the standings after the first round of matches.
Despite the wholesale changes this was a strong Liverpool XI, featuring Virgil Van Dijk as captain and summer signing Gravenberch, who made his full debut following his £34.3 million move from Bayern Munich.
Liverpool's bench was also packed with first-team talent, including Dominik Szoboszlai, Alexis Mac Allister and Salah, in case of emergencies.
Klopp indeed had to break the glass and bring on those three stars after an unimpressive early showing in which Liverpool were stunned by Flecker's opener.
Sascha Horvath's corner found Flecker unmarked on the edge of the box, before Austrian took a touch and smashed the ball into the far corner.
Having fallen behind for the fourth time in five games this season, Liverpool responded and should have equalised before half time when Nunez headed towards goal from just three yards out, only to see LASK keeper Tobias Lawal claw the ball away on the goal line.
They were otherwise held at arm's length by LASK in the opening 45, with Ibrahima Konate's booking for arguing with the referee shortly before the interval testament to the Reds' struggles.
Florian Flecker celebrated his stunning opener against Liverpool with an equally spectacular somersault
Liverpool raised their game and led 10 minutes after the break when Diaz was caught late by LASK defender Philipp Ziereis, with Nunez slamming the spot kick into the bottom corner.
Diaz and Gravenberch then combined with a rare moment of Premier League quality, the Dutchman firing in a low cross from the right which the Colombian lashed past Lawal.
LASK, though, could easily have been back before the Reds took the lead as they created chances of their own which forced Liverpool keeper Caoimhin Kelleher, making a rare start in place of Alisson, in to two fine saves.
The Irishman plunged low to his right to keep out Elias Havel's effort from distance, before saving with his left leg from Marin Ljubicic after a goalmouth scramble.
Gravenberch suffered a knock with 15 minutes to go and was replaced by Salah, who finished coolly through Lawal's legs in the closing stages.
It concluded a performance in which Liverpool fought through rough patches to start their campaign in the Europa League - the only major trophy Klopp has coached in, but not won - with three points.
The victory broke a tie for the most European victories managing Liverpool with Rafa Benitez, who won 49 games between 2004 and 2010.
This was Klopp's 82nd European game at the club, while Benitez managed three more.
-- Courtesy of BBC Sport