The Scousers defeated FC Gomel 1-0 to give new boss Brendan Rodgers a victory in his first competitive match in charge.
The Reds were given a tough workout in Belarus in the first leg of the Europa League third qualifying round tie but they came away with a narrow win thanks to a rare Stewart Downing goal.
The England midfielder scored only his third goal in Liverpool colours in the 67th minute when he cut in from the right and unleashed a 25-yard left-foot shot which skidded past Vladimir Bushma in the home goal.
Gomel, the Belarusian Cup winners last year, had gone close to opening the scoring a few minutes earlier when Sergei Kozeka rattled the upright from long range and Dmitry Platonov sliced wide with the goal gaping after the ball had rebounded into his path.
The home side had the better of the first half as Igor Voronkov failed to direct a close-range header on target and Aleksandr Alumona dragged a shot wide before twice being denied by Reds keeper Brad Jones.
Dmitry Platonov also headed wide and Sergey Matveichik fired over, while Glen Johnson headed wide in a rare Liverpool first-half attack and Fabio Borini sent an angled shot straight at Bushma after the break.
Rodgers was delighted with his side's desire and committment in Belarus.
"It will take time until Liverpool start playing the way Swansea did last season (tactically)," he said. "Actually the way you play depends on the players you have.
"As for today I saw two things which were very important for me, I saw desire and commitment. I am very grateful to my players for a good game.
"Obviously it was a serious test of the players' fitness and I am really happy with the result."
A deflected injury-time strike denied Dundee United a memorable Europa League victory against Dinamo Moscow as the Russians twice fought back in an absorbing 2-2 draw at Tannadice.
Keith Watson looked to have earned United the advantage in the first leg of the third qualifying round after he bundled home from close range - but Alexander Kokorin silenced the home crowd with a 25-yard strike that looped into the net off a United defender.
United had taken the lead against the run of play in the 37th minute when Willo Flood drove home from 16 yards and they created a series of chances before Igor Semshov headed the Russians level five minutes after the break.
United debutant Radoslaw Cierzniak produced a string of excellent saves but the home side kept coming back and Watson struck in the 75th minute.
United suffered narrow defeats to AEK Athens and Polish side Slask Wroclaw in their previous two campaigns and they have given themselves a fighting chance of going one better, although the second away goal leaves Dinamo favourites.
St Patrick's are staring at Europa League elimination after suffering a 3-0 defeat at home to Hannover in the third qualifying round first leg.
The Dublin side, who beat Bosnia's Siroki Brijeg in the last round, were up against it almost immediately after Denmark international Leon Andreasen put Hannover in front in the sixth minute.
Christian Pander doubled the visitors' lead in the 68th minute while Didier Ya Konan sealed an impressive win for the Bundesliga outfit 10 minutes from time.