Liverpool were tested early in the first half. A powerful header from Dani Olmo gave the visitors an early scare. However, Jurgen Klopp's men took control of the game from then on, dictating the play.
Leipzig were well drilled, pressing Liverpool's defence high up the pitch but they lacked stability at the back, which resulted in Salah and Mane breaking through the backline on a number of occasions.
Despite dominating 59% of the first-half possession, Liverpool went into the break with nothing to show for it.
The second half began the way the first had finished. Liverpool commanded the possession but it would be opposition position mistakes that would provide them with their goals.
A misplaced back pass from Marcel Sabitzer set Salah through on goal. Without hesitation the Egyptian drilled a shot to the right of Peter Gulacsi sending him the wrong way and opening the scoring for the Reds.
Liverpool's second would once again come from a mistake from the Leipzig defence. This time it was Nordi Mukiele who failed to connect with a falling through ball. The mistake set Mane through on goal where he mirrored Salah's finish, drilling low to Gulacsi's left and doubling the lead.
It was a commanding win for the Reds and one that should boost the morale of the squad ahead of their Merseyside derby clash with Everton at the weekend.
Liverpool made their intentions clear from the beginning of the match. With Leipzig pressing high up the pitch their back three were left exposed, allowing the likes of Mane and Salah to get in behind.
The off-ball runs from Liverpool's attacking trio unlocked space in behind Leipzig's defensive line and have been something the Reds have been missing in recent matches.
These runs along with outstanding passing from Thiago and Curtis Jones gave Liverpool an attacking threat all evening. The 20-year-old has improved tremendously in the last 18 months, from being selected for domestic cup matches to now starting over the likes of Xherdan Shaqiri and Alex Oxlade-Chamberlain. The midfielder is coming into his own showing off his silky dribbling and range of passing.
The threatening attacks almost amounted to a goal for Liverpool in the opening 45. Roberto Firmino had his goal disallowed when the ball crept over the touchline just before Mane stabbed a ball across the box to Firmino.
Andy Robertson would be the next to try and break the deadlock but to no success. After the Leipzig 'keeper intercepted a through ball his clearance fell to the feet of Liverpool's left-back on the halfway line. With a touch and shot, Robertson skimmed the roof of the net leaving Gulacsi scrambling in his goal.
A clean sheet at last
It had been 13 games since the Reds had won a match with a clean sheet; the match being the 7-0 win against Crystal Palace back in December.
Similar to that game, Liverpool didn't face many attacks against Leipzig but when called upon the defence performed resolutely. After their mistakes against Leicester City, Jordan Henderson and Ozan Kabak looked reformed, winning their individual duals whilst keeping possession well and distributing the ball positively.
Alisson made some outstanding saves denying Olmo and Christopher Nkunku from close range. However, the Brazilian still showed signs of lapse when playing the ball out from the back. His kicks ended up at the feet of Leipzig players on a handful of occasions which luckily did not result in any damage to the scoreline.
The same could not be said for the Leipzig's defence which looked panicked at the back all evening. Newly announced Bayern Munich defender Dayot Upamecano was the main culprit, giving away possession to Liverpool's attackers in dangerous positions throughout the match.
These mistakes led to Liverpool's goals and if they are not fixed then Klopp's men will look to exploit them in the return fixture.
Seeing the game out
Against Leicester last weekend, the Reds lost confidence as soon as they conceded one and lost the game in embarrassing style.
Fortunately, it was quite the opposite against Leipzig. Calm and composed in possession, Liverpool dictated the tempo of the game from start to finish.
In the latter stages of the match, Klopp's men remained confident and wore out their opposition who had been relentless in pressing from the first whistle.
To see Liverpool play out a match without panicking or trying too much in the dying minutes of a game is a positive sign. It shows they are still confident in their own game and this may be a sign of improvement to a drained squad.