Jürgen Klopp's players lost all three away games in last season's group stage, and the title holders were beaten at Napoli to start their new campaign.
With six points from three matches, Liverpool remained in second place in Group E — one point behind Napoli, which beat Salzburg 3-2.
"There were good moments. The start was brilliant, kept it kind of fluent and gave them a lot of questions with the flexible formation," Klopp said. "But then we started, I'm not sure why, to lose patience and a lot of easy balls. All of the counter-attacks they had were when we lost an easy ball. But all four goals were brilliant. We could have scored more, but job done."
Klopp made five changes to the team that was held 1-1 by Manchester United in Premier League over the weekend, including the fit again Mohamed Salah, who returned from an injury.
It was Oxlade-Chamberlain's first double since he scored twice for Arsenal in a 2-0 League Cup win over Reading nearly three years ago.
Sadio Mane made it 3-0 with a chipped ball over 'keeper Gaetan Coucke in the 77th minute and Salah completed the rout 10 minutes later after using quick footwork to create space inside the area.
The hosts were caught cold early after giving the ball away in their own half. Liverpool put together a string of passes and Fabinho delivered a precise assist for Oxlade-Chamberlain, who scored in the second minute with a low shot from outside the box on his first Champions League appearance in 18 months.
Oxlade-Chamberlain, who was side-lined for almost all of last season with a knee injury he picked up in the semi-finals of the 2018 Champions League, doubled his tally in the 57th minute from Roberto Firmino's pass when he fired a superb curling shot with the outside of his right foot just under the crossbar.
Stephen Odey pulled one back for Genk with two minutes left.
Playing high on the pitch, Liverpool controlled ball possession but the Reds' defence was often exposed on the break.
Genk had a couple of good chances within the first 10 minutes as Ally Samatta missed the target and Liverpool keeper Alisson made a fine save to deny Paul Onuachu's effort. Samatta thought he snatched an equalizer near the half-hour mark with a header at the far post, but his goal was eventually cancelled after a VAR check, for an offside in the build-up.