The England midfielder's last appearance in the competition came in the 2018 semi-final, when he suffered a serious knee injury that decimated a year of his career.
Recalled to Jurgen Klopp's midfield alongside Naby Keita, Oxlade-Chamberlain's second-minute opener took a slight deflection but a majestic second before the hour was emphatically all his own work.
Substitute Stephen Odey swivelled to finish in the 88th minute - a deserved reward for Genk's attacking endeavours - although Sadio Mane and the returning Mohamed Salah had put the result way beyond doubt by that stage.
Genk quickly faced a tall order when Oxlade-Chamberlain was given ample room 25 yards from goal to line up a low drive that deflected past the helpless Gaetan Coucke.
Bryan Heynen snatched at a chance to equaliser before Alisson saved from Paul Onuachu – the Liverpool defence undone by two long balls over the top in quick succession.
The same ploy looked like it might bear fruit in the 20th minute, only for Junya Ito to lose his footing inside the visitors' box.
Mane was denied by a fine Coucke save on the end of a return rabona flick from Roberto Firmino before Liverpool were granted a VAR reprieve – Ito shown to be fractionally offside before his inviting cross was powered home by Mbwana Samatta.
Samatta was unable to slide in a low 40th-minute delivery from Ito as Liverpool again looked defensively vulnerable.
The hosts' pacey, direct approach once more looked dangerous after the restart, meaning Oxlade-Chamberlain's immaculate second gave Klopp's men much-needed breathing space.
Firmino latched on to Naby Keita's chipped pass and laid it back to the opening goalscorer outside the area, where a sublime strike with the outside of his right boot sent the ball whistling past Coucke and into the net via the underside of the crossbar.
Salah teed up on Mane for a cute 77th-minute finish before dribbling into space and slotting past Coucke three minutes from time for another quality Liverpool goal as they stayed a point behind Group E leaders Napoli, though Odey's late goal spoiled their clean sheet.
What does it mean? Klopp's men unbowed after tough away assignments
Liverpool were far from their fluent best before Adam Lallana snatched a late draw at Manchester United on Sunday and Genk made much of the running here. The Premier League leaders were again able to offset signs of weakness with rugged grit and determination – qualities best typified by James Milner getting to work in both full-back positions.
Year of the Ox?
It has been a long, gruelling road back to this point for Oxlade-Chamberlain and his was a display to warm the coldest heart. The former Arsenal man provides a goal threat from midfield Liverpool often lack, although few of his efforts from this point will match his stunning 57th-minute intervention.
Genk attack leaves Robertson squashed
The lively and erratic Ito delighted and frustrated the Genk faithful in equal measure but he subjected Reds left-back Andy Robertson to an unusually torrid outing. Klopp's decision to send on Joe Gomez in his place and push Milner to left-back in the 63rd minute spoke volumes.
Liverpool host Tottenham on Sunday, with Mauricio Pochettino's men looking to build on a morale-boosting 5-0 win over Red Star Belgrade. Genk host Cercle Brugge a day earlier as they try to improve upon their sixth position in the Belgian First Division.