The Reds were set on their way to an ultimately comfortable Champions League victory in Belgium as early as the second minute, when Oxlade-Chamberlain found the bottom corner from distance.
Watch: Genk 1-4 Liverpool in 90 seconds
The No.15’s sublime second just before the hour tightened the European champions’ grip on the Group E clash, before Mane rounded off a fine team move 13 minutes from time.
Mane, who had been assisted by Salah, then returned the favour for the Egyptian to make it 4-0, with Stephen Odey subsequently grabbing a consolation for Genk.
Napoli’s 3-2 win over Salzburg means Liverpool remain second in the pool, sitting a point behind Carlo Ancelotti’s team and three above the Austrian club.
Team news
Jürgen Klopp freshened up Liverpool’s side by making five changes to his line-up, with Oxlade-Chamberlain and Salah joined in the starting XI by Naby Keita, Dejan Lovren and James Milner.
First half
Oxlade-Chamberlain’s opener was fashioned via the Reds’ apparent early ploy to attack down their left.
The ball was worked inside from that flank to Fabinho, who punched a pass into the feet of the Oxlade-Chamberlain. The midfielder’s first touch enabled him to find a favourable angle for a shot, and he duly located the bottom corner from 25 yards with the assistance of a minor deflection.
Liverpool’s early dominance could have been undone had Genk been more clinical on the counter, however.
First, Mbwana Samatta drove wide, before Milner, with a last-ditch tackle, and Alisson Becker, who mopped up alertly, combined to thwart Paul Onuachu.
The Champions League holders continued to showcase flashes of their quality, though, with one outrageous flick from Roberto Firmino midway through the half drawing gasps from the crowd.
That piece of ingenuity created a chance for Mane too, but the No.10’s low effort was repelled by Gaetan Coucke.
Genk thought they were level soon after, when Samatta planted a firm header beyond Alisson, but Junya Ito was adjudged to have been offside while delivering his cross from the right – a decision that was upheld by a VAR review.
Salah brushed the top of the net with a free-kick and then miscued his header when unmarked after Lovren had leapt to meet Milner’s corner, and the final attempt of the half was Fabinho’s snap-shot volley that was gathered by Coucke.
Second half
Genk made an enterprising start to the second period, winning a couple of corners in quick succession, but their efforts were undone by Oxlade-Chamberlain’s jaw-dropping moment of class.
Patient build-up eventually freed Firmino in the inside-left channel of the box, with the No.9 teeing up the midfielder on the edge of the area.
What followed was magnificent, Oxlade-Chamberlain sending an impudent, outside-of-the-right-boot finish in off the crossbar from 20 yards.
Razor-sharp link-up between Firmino, Salah and Mane, who clipped a fine shot over the onrushing Coucke, further extended Liverpool’s advantage after substitute Georginio Wijnaldum had regained possession high up the pitch.
And Mane quickly turned provider for Salah, the No.11 rolling Carlos Cuesta and displaying brilliant footwork prior to easing a low strike into the bottom corner.
Genk immediately went up the other end and pulled one back through Odey, but a professional display meant the Reds already had their second successive Group E win wrapped up.