However, they were stunned by hosts LASK in the first 15 minutes despite starting well.
Sascha Horvath crossed from a corner, and Florian Flecker controlled before drilling his effort beyond Caoimhin Kelleher to give his side the lead in Linz.
Liverpool came close to netting an equaliser as Harvey Elliott’s cross found Darwin Nunez, but the Uruguayan attacker could not keep his header down.
Nunez had another magnificent chance with a header, but LASK keeper Tobias Lawal kept the effort out as the home side took their lead into the half-time break.
Liverpool piled the pressure in the second half. After brilliant interplay between Elliott and Stefan Bajcetic, Luis Diaz’s shot went wide, but Phillipp Ziereis’ lunging tackle took the Colombian down.
The referee gave a penalty, and Nunez stepped up to convert it.
However, LASK did not let their heads drop and came flying forward. Elias Havel found a decent angle and struck a good shot, forcing Kelleher to make the save.
From a corner, the ball fell to striker Marin Ljubicic, but Kelleher pulled off an incredible save with his boot before the linesman raised his flag.
Liverpool restarted and scored immediately. Elliott released the impressive Ryan Gravenberch down the flank. The Dutchman crossed and Luis Diaz finished to give his side the lead after 63 minutes.
Mohamed Salah came on and tormented LASK. He had an effort cleared off the line before he netted late in the game to secure the victory for the Reds.
The Merseyside are now in control in Group E ahead of Union St Gilloise and Toulouse. They went down for the fourth time in the last five games, but found a way to secure the victory.
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