Jurgen Klopp's side were totally in command of their last-16, first-leg tie in Portugal, where first-half goals from Sane and Mohamed Salah handed them a 2-0 lead at the break, before Roberto Firmino and the Senegal global completed the rout in the second half.
The win came in Liverpool's first Champions League knockout in nine years, the last being the second leg loss to Chelsea in the quarter-finals of the competition in 2009.
They need that success to translate over to the Champions League, though, starting with their Round of 16 tie against Porto.
Mohamed Salah and Roberto Firmino also got on the scoresheet as Liverpool easily brushed aside their Portuguese opposition.
"It was important to play as a team and we did, we played great football from the beginning", Mane told BT Sport.
Sadio Mane was the hero with a superb hat-trick as the Reds ran riot in Portugal, but Klopp was also quick to praise his defence, the subject of plenty criticism in recent times. Or can PSG comeback in the second leg in France in two week time? "It's easy to score eight past Dijon and score four or five in the [French] league, but we have to be decisive in games like this".
And the Liverpool fans couldn't control themselves.
It was soon 2-0 as James Milner hit the post with an effort from the left corner of the box and the rebound fell for Salah, who coolly finished, chipping over the keeper then heading down to himself to tap in from close range. If we are in the final then I'll say we will try to win it but that is too far off.
Salah's 29th-minute goal arguably upstaged Mane's treble as the Egyptian took his tally to 30 goals in all competitions this season. That's important tonight. We spoke a lot about Mo in the last few weeks, completely deserved. "The third one pleased me the most because it was a nice strike".
"The way that he sort of controlled it with his head, sort of juggled the ball and converted with so much time, the time that the very best players have". Without the team we are nothing, they do a great job behind us.