Mohamed Salah showed superb awareness to quickly latch on to a loose back-pass before finishing neatly.
Sadio Mane was then equally alert to take advantage of a fluffed attempted clearance from Nordi Mukiele to charge through and side-foot past Peter Gulacsi.
In what had been a fast-paced match between two high-pressing sides, the Reds’ greater experience and quality had started to tell before the break.
However, they were fortunate not to fall behind inside five minutes when Dani Olmo’s header from an Angelino cross struck the base of the post.
Having managed the game well after going 2-0 up, Jurgen Klopp’s side will defend their advantage at Anfield on 10 March, when they will be firm favourites to go one round further than they managed last season.
Liverpool take first step on road to recovery
In the gap between their final group game and this, Liverpool’s season has imploded somewhat. Their long unbeaten domestic home record has come to an end, as has their challenge for this season’s FA Cup, and their last three league matches have ended in defeat – leading boss Jurgen Klopp to wave the white flag on their title defence.
Tuesday night, though, was a much-needed reminder of the qualities that drove them to that league win and the Champions League trophy the season before it.
Having survived Olmo’s early header, they grew into the game, wresting control away from Leipzig before shellshocking opponents who reached the semi-finals of this competition last year, saw off Manchester United in this season’s group stage and currently sit second in the Bundesliga.
The game was billed as master versus apprentice, one of Germany’s finest managerial exports of the modern era in Klopp up against the country’s – possibly the continent’s – brightest young thing in Julian Nagelsmann.
It showcased a shared DNA between two sides intent on playing on the front foot in enemy territory. Players charged forward when they had the ball and harried in a high press when they did not. Fire fought with fire.
It made for a frenetic and fascinating spectacle, which tested even the most lauded of defenders and forced them into errors, chief among them Bayern Munich-bound Dayot Upamecano, who repeatedly gave the ball away as Mane and Salah buzzed around him.
Despite having the better of the first half, applying the more consistent pressure and producing the greater number of chances, Klopp’s side were unable to get their noses in front.
In the space of 30 seconds, Salah saw an effort blocked by Gulacsi after he had darted in behind the Leipzig defence before Roberto Firmino fired into the side-netting from an angle.
Mane headed over another opportunity, Robertson was inches away from the beating the back-peddling Gulacsi with an effort from near halfway and Firmino had the ball in the net but only after the ball had gone out of play before being hooked back into the middle.
However, these were all forgotten before the hour mark was reached as the German side snapped and Salah and Mane took advantage.
Leipzig never fully recovered, their only real chance of salvaging anything to take to Merseyside being a shot from an angle that substitute Hwang Hee-chan pulled wide in added time.
It represents a big foot forward towards the quarter-finals for the Reds and a first step on the road to potentially reigniting their flagging season.