Leipzig had reached this stage by eliminating Manchester United, but a side that has the best home record in this season’s Bundesliga looked uncomfortable away from their Red Bull Arena fortress.
They started well, putting Liverpool on notice early on as ex-Manchester City full-back Angelino crossed to Olmo, whose low diving header evaded Alisson Becker at his near post and clattered the inside of the goal frame.
But Liverpool became increasingly assured as the half went on, with their vaunted front three creating a number of problems for a Leipzig defence whose high line left them repeatedly susceptible.
Salah collected a piercing pass on the inside right but was unable to lift over goalkeeper Peter Gulacsi, who had to bail out his defence to reach Andy Robertson’s ball over the top before the onrushing Mane.
However, Gulacsi, who spent six years at Anfield without making a first-team appearance, had to scramble back and was nearly caught out by an alert Robertson, whose attempt from almost halfway landed on the roof of the net.
Roberto Firmino was denied the opener after he headed into an empty net, because the ball had gone out of play before Mane hooked in from the byline, sparing Dayot Upamecano for another moment of sloppiness.
For all of Liverpool’s pressure, they were almost caught out either side of the interval as Olmo found space in the area but missed the target with his unmarked header before Alisson produced a fine save to deny Christopher Nkunku.
Alisson has made high-profile blunders in Liverpool’s most recent two defeats but after Nkunku had got goal side of Jordan Henderson to take Olmo’s through ball, the goalkeeper was able to claw away the effort.
It would prove a valuable intervention as Liverpool emphatically seized control of the tie before the hour with quickfire goals from Salah and Mane – although both came courtesy of dreadful errors from their opponents.
There seemed to be little danger when the ball fell to Sabitzer but the Leipzig captain was wayward with his pass to last man Lukas Klostermann, allowing Salah to steal in and finish clinically in the 53rd minute.
Mane was gifted another one-on-one opportunity five minutes later and, like Salah before him, he made no mistake, put through after Mukiele slipped while attempting to cut out a long clearance from Curtis Jones.
Angelino could have pulled one back but fired wastefully across goal when in acres of space in the area. Hwang shot wide in the dying moments after getting in behind the defence and lifting over Alisson.
Liverpool therefore kept a priceless two-goal lead ahead of the return leg, scheduled to take place at Anfield on March 10, while their confidence is bound to have improved ahead of this weekend’s Merseyside derby.