Ryan Gravenberch opened his account for the Reds on the stroke of half time thanks to a goalkeeping gaffe, and while Jurgen Klopp's side were not at the races for most of the second half, an injury-time strike from Diogo Jota sealed the deal.
In a first half that was incredibly frustrating for Liverpool - for the most part at least - the hosts were left to rue a period of wastefulness, starting with Mohamed Salah failing to beat Anthony Moris from a good position after Jarell Quansah's interception sailed through to him.
It was then Darwin Nunez's turn to fall foul of the rulebook and his own profligacy, as the Uruguayan crashed home in the 10th minute after Moris parried Gravenberch's drive into his path, but the offside flag spared Union SG's blushes.
Nunez was presented with a gilt-edged chance to propel Liverpool ahead just 17 minutes later, as Salah squared for his teammate to ostensibly find the net with a simple tap-in from a few yards out, but the South American dragged his shot wide of the far post.
However, while Moris got away with deflecting one strike back into the danger area, the Union SG goalkeeper was not so fortuitous a second time around, as Liverpool finally broke the deadlock one minute before the half-time whistle sounded.
A rapid Liverpool break ended with Trent Alexander-Arnold having a crack from distance, and Moris spilled the right-back's effort into the path of Gravenberch, who pounced to open his account for the Merseyside giants.
A raft of half-time changes from the Reds saw Alexis Mac Allister introduced for an on-field reunion with his brother and Union SG defender Kevin Mac Allister, who almost had an equaliser for his side to celebrate in the 50th minute as Alisson made a pig's ear of a corner, but the alert Quansah hacked the ball away from the open net.
Down the other end of the field, Moris redeemed himself for his first-half howler with a fine reach to tip Jota's header over the top in the 52nd minute, before also sticking an arm up to push Gravenberch's curler to safety just after the hour mark.
Still, the hosts' position remained vulnerable, and Alexander Blessin's side began to take a few more risks in search of a leveller, but Liverpool stood firm as the visitors committed more bodies forward.
An increasingly rare Liverpool attack down the Kop end saw Luis Diaz curl a low strike onto the post in the 80th minute, although he was in an offside position when Ibrahima Konate played a beautiful long pass to him, before the Union SG crowd appealed for a penalty for Quansah's one-handed shove on Kevin Rodriguez - their pleas fell on deaf ears.
The visitors were given five minutes of added time to ruin the Liverpool party, but Klopp's side would extinguish any Union SG hopes of a fightback in the second, as Diaz forced Noah Sadiki into a mistake and released Jota to fire home across Moris's goal from the left-hand side of the box.
Liverpool would not march to victory without one final scare, as a lapse in concentration presented Mohammed Amoura with a one-on-one against Alisson, but the Brazilian came out on top as the Group E leaders moved onto six points from a possible six.
Liverpool's final game before the international break comes away to Brighton & Hove Albion on Sunday, while Union SG have a Jupiler Pro League trip to Sint-Truiden on the agenda this weekend.