Never in Liverpool’s history had they lost a European tie after winning the first leg away from home but Ousmane Dembele’s early strike brought the French champions level at 1-1 on aggregate.
PSG were then perfect with their four spot-kicks, while Gianluigi Donnarumma saved from Darwin Nunez and Curtis Jones.
Luis Enrique said before the game that the winners would progress to the final and PSG will be supremely confident of reaching the semi-finals at least as they will face Aston Villa or Club Brugge in the last eight.
A famous victory for the Parisians was deserved over the tie after they dominated the first leg only to be denied by the brilliance of Alisson Becker in the Liverpool goal before Harvey Elliott snatched a late winner.
The roles were reversed early on and Liverpool were made to rue not making more of their chances.
Brilliant play by Alexis Mac Allister should have created the opener on four minutes when the Argentine’s cross picked out Mohamed Salah, whose goalbound effort was deflected over by Nuno Mendes.
Salah sliced wide another good chance on one of the few occasions he got the better of the determined Mendes.
PSG took 10 minutes to ride out the storm from the home side before finding their feet.
The visitors took the lead on 12 minutes when Ibrahima Konate could only turn Bradley Barcola’s cross beyond Alisson to leave Dembele with a simple finish.
The French international has now scored 23 goals in his last 17 games and could have had more before the end of the night.
Konate nearly made amends immediately when his low shot was turned behind by Donnarumma.
However, it was PSG who had the better openings to extend their advantage before half-time. Twice Alisson had to be at his sharpest to race off his line and save one-on-ones against Barcola and Dembele.
PSG beat Liverpool to the signing of Khvicha Kvaratskhelia in January and the Georgian proved a constant menace to the Reds’ defence.
Kvaratskhelia’s shot from Dembele’s cross was arrowing towards the top corner until a deflection from Ryan Gravenberch took it just over the crossbar.
Improved second-half performances have been a common theme of Arne Slot’s first season at Liverpool and the hosts came roaring out of the traps after the break.
Dominik Szoboszlai had a goal ruled out for offside against Luis Diaz earlier in the move.
Szoboszlai then had a goalbound effort blocked by Willian Pacho before Donnarumma made a vital stop from Diaz from the resulting corner.
Liverpool’s momentum could have been halted by the loss of Trent Alexander-Arnold to a knee injury after the right-back landed awkwardly.
But his replacement, Jarell Quansah, was inches away from winning the tie when his header came back off the inside of the post.
PSG barely threatened during the second half of the 90 minutes but Liverpool were left hanging on for penalties in extra time.
Lucas Beraldo’s header and Desire Doue’s shot flew just wide before more Alisson heroics kept PSG at bay with a stunning stop to turn Dembele’s curling effort round the post.
However, Alisson could not save them in the shootout as it was the other goalkeeper who was the hero.
Donnarumma produced great saves to deny Nunez and Jones, either side of four confident strikes from Vitinha, Goncalo Ramos, Dembele and Doue to send PSG into the last eight.
In the day’s other results, Inter Milan are set to face Bayern Munich in the quarter-finals after seeing off Feyenoord 2-1 to complete a comfortable 4-1 aggregate victory. (AFP)
