The holders are one point ahead of Napoli, who could do no better than a 1-1 draw in their match Tuesday against third-place Red Bull Salzburg, reports Efe news.
Genk, still looking for their first-ever win after 15 Champions League matches, are the weakest club in the group and nobody was surprised to see Liverpool coach Jurgen Klopp leave Roberto Firmino, Sadio Mane and Andrew Robertson on the bench to rest them on Tuesday ahead of Sunday's Premier League showdown with Manchester City.
Even so, the Reds dominated the early going in front of their supporters at Anfield and took the lead in the 14th minute on a goal by Georginio Wijnaldum.
The Dutch international scored with an assist from James Milner, Liverpool's best player of the night, who came close to making it 2-0 seven minutes later.
Instead, the next goal was to come from Genk, as an unmarked Mbwana Samatta headed-in a corner in the 41st minute to bring his side level against a seemingly distracted Liverpool.
Samatta inspired his teammates and Genk midfielder Bryan Heynen didn't miss by much two minutes later.
The visitors had another chance to go ahead early in the second half, but forward Junya Ito opted to pass instead of taking the high-percentage shot.
Liverpool's Alex Oxlade-Chamberlain didn't hesitate when opportunity knocked in the 53rd minute, converting a pass from Mohamed Salah to put the hosts up 2-1.
( With inputs from IANS )