Liverpool took an early lead through Georginio Wijnaldum before Ally Mbwana Samata put Genk level on 41 minutes. Alex Oxlade-Chamberlain scored the winning goal for Liverpool 8 minutes into the second half.
Liverpool the lead in the 14th minute with a close-range effort from Wijnaldum after James Milner’s low cross from the left bounced off Sebastien Dewaest and fell to the Dutchman who scooped home.
The English side had several chances to extend their lead but lacked the killer touch in front of goal with Mohamed Salah below his usual level of sharpness, wasting a good chance by blasting wide after another slip-up from Dewaest.
Against the run of play, Liverpool were pegged back four minutes before the break, when KRC Genk’s Tanzanian striker Samatta met Bryan Heynen’s corner with a superbly-angled near-post header.
Liverpool were on top again in the second half with the Belgian Divock Origi missing a good chance and Virgil van Dijk flashing a header just wide.
Oxlade-Chamberlain the home side’s advantage in the 53rd, collecting a neat pass from Salah on the edge of the box and beating the KRC Genk keeper Gaetan Coucke with a low drive on the turn.
Liverpool were comfortably in control from then on but keeper Alisson Becker had to be alert nine minutes from the end to push out a dangerous shot from the impressive Heynen.
Genk remain without a win in the group stage and are bottom with one point. Nevertherless, the Belgian Champions can take some heart from a performance which caused Liverpool plenty of frustration.
Speaking after the game the Genk defender Casper De Norre told the press agency Reuters that “With a bit of luck, it could even have become 2-2. Conceding only two goals at Anfield is not too bad. You know you’re put under pressure that is something you can never avoid here.”