Jurgen Klopp's side travelled to the Puskas Arena smarting from suffering a sixth consecutive home defeat in the Premier League at the hands of Fulham, but there was a sense of deja vu as they moved into the last right.
At the same venue where Salah and Mane gave the Reds a 2-0 first-leg advantage in what was Leipzig's home leg, the Reds forwards were on target once again to seal a 4-0 aggregate win.
Liverpool wasted a host of chances, but Alexander Sorloth hitting the crossbar was the closest Leipzig came as last season's semi-finalists bowed out.
Mane volleyed over amid a bright start for the Premier League champions, who had Fabinho back in midfield, before Dani Olmo was denied by the alert Alisson at the other end.
Liverpool did not resemble a side devoid of confidence and Peter Gulacsi showed sharp reactions to palm Diogo Jota's powerful header from a corner beyond the crossbar.
Salah wasted a great opportunity to open the scoring when Thiago's sublime pass sent him clear, but the forward shot straight at Gulacsi before he and Mane were unable to finish following up.
Emil Forsberg dragged a shot tamely wide and Jota spurned a golden chance when he broke through the middle but failed to beat Gulacsi, with the Portugal international then only finding the side-netting when he should have scored following an Dayot Upamecano error on the stroke of half-time.
Jota allowed Gulacsi to save all too easily yet again 10 minutes into the second half and Salah blazed the rebound over the bar as the breakthrough remained elusive.
Sorloth, introduced as a half-time substitute, headed Hee-Chan Hwang's inviting cross against the bar midway through the second half, with Leipzig battling to stay in the competition.
Liverpool had one foot in the last eight 20 minutes from time, though - Jota setting up Salah in a rapid attack and the Egypt international cutting into the penalty area before finding the bottom corner with his left foot.
Divock Origi then whipped in a brilliant cross for Mane to tuck home a second as Klopp's men breezed through.