Elsewhere in the early matches in the first round of the group stage of the second-tier European competition, Romelu Lukaku earned last year's beaten finalists Roma a 2-1 victory at Sheriff.
Bayer Leverkusen crushed Swedes Hacken 4-0, Rennes cruised past Maccabi Haifa 3-0 in France, Spaniards Villarreal lost 2-0 to Panathinakos in Greece.
Liverpool, playing in the Europa League for the first time since finishing runners up in 2016, changed their entire starting 11 away to LASK and quickly fell behind.
In the 14th minute, Sascha Horvath lobbed a corner to Florian Flecker just outside the Liverpool box. The Austrian midfielder controlled the dropping ball before smashing a rocket shot past Caoimhin Kelleher in the Liverpool goal.
Darwin Nunez levelled with a 55th-minute penalty kick after Luis Diaz was brought down.
After 61 minutes, Liverpool coach Jurgen Klopp started to replace teenagers with regular starters.
Within two minutes they were ahead when Ryan Gravenberch burst down the right and crossed low for Diaz to score from close range.
Two minutes from the end Mohamed Salah, a second-half replacement, poked home a third.
In Moldova, Leandro Paredes scored four minutes into first-half injury time to give Roma the lead away to Sheriff.
Colombian Cristian Tovar levelled but Romelu Lukaku gave Jose Mourinho's team a narrow victory.
In the Conference League, Aston Villa, returning to European competition after a 12-year gap, fell behind in the third minute to Legia Warsaw and despite equalising twice lost 3-2.
Pawel Wszolek met a cross from the left with a sliding volley to put Legia ahead.
Jhon Duran levelled after six minutes heading in a rebound after a Nicolo Zaniolo shot.
But the Poles regained the lead after 25 minutes when Wszolek crossed from the right and Ernest Muci scored on the half volley.
Lucas Digne levelled with a deflected shot from the edge of the box after 38 minutes but Muci settled the game with a low shot after 51 minutes.
Croatians Zrinjski Mostar came from three goals down to breat visiting AZ Alkmaar 4-3.
Myron Van Brederode, Sven Mijnans and Dani de Wit put the Dutch side three goals up by half-time.
Zvonimir Kozulj started the comeback afrer 48 minutes and ended it in the 81st minute, sandwiching goals from Josip Corluka and Aldin Hrvanovic.
