James Milner, Divock Origi, Alex Oxlade-Chamberlain, Joe Gomez, Fabinho and Naby Keita came in for Andy Robertson, Sadio Mane, Roberto Firmino, Dejan Lovren, Jordan Henderson and Adam Lallana.
The reigning champions went into the game boasting a 23-game unbeaten run in European, stretching back over five years to October 2014.
And, after dominating possession in the early exchanges, the Reds took the lead in the 14th minute when Georginio Wijnaldum lifted the ball over keeper Gaetan Coucke after Sebastian Dewaest failed to clear a Milner cross into the area.
Milner then tried his luck on 21 minutes with a volley from the top of the area but Coucke got down low to make a good save.
Genk’s only effort of note came just after the half-hour when Alisson got down and made a fairly routine save to deny Joakim Maehle with a cross-shot.
Liverpool should have doubled their lead on 36 minutes when Mohamed Salah got the better of Dewaest in the area and whipped a low ball across the area which Origi just couldn’t reach at the back post.
The hosts were made to pay for their profligacy when Genk equalized with one of their rare attacks on goal.
Bryan Heynen played the corner into the area which just cleared the head of Oxlade-Chamberlain and fell kindly to Mbwana Samatta who made no mistake with a powerful header past Alisson.
Origi looked to put Liverpool back in front against his former club on 49 minutes as he curled a shot at goal which was curling into the corner until Coucke scrambled across his area to push it wide.
Liverpool did get back in front on 53 minutes when Salah teed up Oxlade-Chamberlain on the edge of the area and he fired a powerful strike on the turn past Coucke.
The Reds continued to dominate and had several chances but couldn’t find a killer second; while Alisson was called on to make a big save on 82 minutes to deny Heynen.
Despite the less than convincing display, Liverpool held out for the win to move top of the group, a point clear of Napoli who were held to a 1-1 draw at home against Salzburg.