Dejan Lovren and Ragnar Klavan came to Liverpool's rescue in stoppage time to secure a 2-1 victory after Burnley had found a late equaliser at Turf Moor on Monday.
With Philippe Coutinho and Mohamed Salah both left out of the squad, Liverpool struggled to find much of an attacking threat and the hosts came closest in the first half when Scott Arfield dragged a shot inches wide of the post.
Sade Mane delivered for Liverpool on the hour mark, using one touch to control Trent Alexander-Arnold's deflected cross into the box before spinning and sending a brilliant blast high into the net.