Liverpool gave their flagging season a timely boost by beating RB Leipzig to reach the Champions League quarter-finals.
The English champions have seen their hopes of retaining the league title they won last June all but ended by a shocking run of recent domestic form, which includes defeat in each of their last six games at Anfield.
But their quest for a second European crown in three seasons remains very much alive thanks to a more confident and composed display in what was technically a home fixture but played at the same venue as the first leg - Budapest's Puskas Arena.
For 70 minutes, Liverpool were extremely wasteful in front of goal, but sealed the tie when Mohamed Salah cut inside and finished low shortly after Leipzig substitute Alexander Sorloth had struck the bar with a header.
Sadio Mane made absolutely sure with a neat finish from Divock Origi's superb cross.
Jurgen Klopp's side will find out their last-eight opponents when the draw is made on Friday, 19 March.
Meanwhile PSG survived a first-half onslaught from Barcelona, where Lionel Messi scored but missed a penalty, to draw 1-1 and progress 5-2 on aggregate in the Champions League.
Ronald Koeman's side needed to score four goals for the tie to swing in their favour in Paris and started like a team capable of producing them with Osama Dembele wasting some glorious early chances.
It was to be PSG who opened the scoring when referee Antony Taylor awarded a penalty via the use of the pitchside monitor to deem that Mauro Icardi had been fouled off the ball by Clement Lenglet. The penalty was dispatched by Kylian Mbappe as he became the youngest player in Champions League history to reach 25 goals in the competition, taking the mantle from Messi.
Despite falling behind, Barcelona remained positive and Messi thrashed home a fabulous strike to inspire some hope but that hope faded away when his penalty was saved by Keylor Navas just before half-time.
The Spanish giants needed three second-half goals to force extra-time but PSG were much better organised after the break and restricted Messi's influence on the game.
This will be the first Champions League season without a Messi or Cristiano Ronaldo side in the quarter-finals since 2004/05.