Goals from Mohamed Salah and Sadio Mane proved decisive for the Premier League side as they ended a run of three straight defeats.
Leipzig's 'home' leg was played in the Hungarian capital because the German government has banned nations with mutant strains of the Coronavirus from entering the country.
The two sides are due to play the return leg at Anfield on March 10, but that game could still yet be moved to a neutral venue too.
So who were the standout performers on the night in Budapest?
✔️ GOOD PERFORMANCE
Mohamed Salah: 7.4
The Egyptian has been criticised for his lack of consistency this season but the Leipzig defence couldn't cope with him on Tuesday.
In the first half, he was denied by an excellent Peter Gulacsi save as he ran through on goal.
And he made no mistake after the break as he capitalised on a dreadful error by Marcel Sabitzer and sent Gulacsi the wrong way with his finish.
Salah had an impressive six touches in the box and was a constant threat with his pace as Leipzig pushed their defence up suicidally high.
Curtis Jones: 7.3
A really mature performance from the young midfielder, who ran the show in the centre of the park alongside Thiago Alcantara and Gini Wijnaldum.
He provided the assist for Mane's goal, and boasted a decent pass completion rate of 85 per cent.
Jones' disruptive work was also impressive as he made five recoveries and two interceptions. With Jordan Henderson in defence, he more than filled in for the captain.
Nordi Mukiele: 4.8
In truth, all of Leipzig's defenders were disastrous - even if they weren't helped by Julian Nagelsmann's decision to push the backline so high up the pitch.
Countless unforced errors were made during the game - including on both of Liverpool's goals.
And the mistake by Mukiele before Sadio Mane's strike will feature in many Christmas blooper videos. He lost the flight of the ball completely in the air on the half-way line, allowing Mane to run through and score.
⭐ STAR MAN
Sadio Mane: 7.5
The Senegalese star returned to his best against Leipzig, having struggled for form for much of 2021.
He finished with aplomb to put Liverpool in a commanding position at 2-0 and, like Salah, his pace was always going to cause trouble against such a high line.
Mane had a game-high four shots, five touches in the box and two dribbles.