Liverpool 4 Borussia Dortmund 3 (Liverpool win 5-4 on aggregate)
Liverpool have form for stunning European comebacks and this will go down alongside the best of them as they snatched an added-time winner to beat Borussia Dortmund in a modern-day Anfield classic.
All hope seemed lost at 2-0 down - 3-1 on aggregate - after 10 minutes and later at 3-1 behind on the night and 4-2 adrift over the two legs, but the hosts dug deep and produced a remarkable turnaround which saw Dejan Lovren score a dramatic winner. A 4-3 win carried Liverpool through 5-4 on aggregate.
Manager Jurgen Klopp's assertion back in November that "we decide when it is over" rang true as his players, while not of the 2005 Istanbul vintage, gave every part of their being to put them into the Europa League semi-finals.
Henrikh Mkhitaryan and Pierre-Emerick Aubameyang put the visitors ahead early on, cancelling out Liverpool's away goal in Germany, before Divock Origi's fourth goal in his last three games offered some hope.
Marco Reus appeared to have extinguished hopes but Mamadou Sakho and Lovren, both central defenders, proved to be the unlikeliest of heroes on a night when Klopp's side came of age.
Dortmund boss Thomas Tuchel had promised to attack with impunity and score "two, three or four goals" and his side carried out his instructions to the letter only to be stung so painfully.
In a blistering start which left their hosts shell-shocked, Dortmund looked to be sending Liverpool out of the competition with the sort of football on which they have made their name.
Liverpool lost possession on their left in Dortmund's half and the German side countered. Aubameyang's volley from Gonzalo Castro's cross was worthy of a goal but Simon Mignolet saved well, only for Mkhitaryan to fire home the rebound.
With the away goal cancelled out, Dortmund went for the jugular, with another swift counter-attack seeing a brilliant pass from Reus in behind Sakho superbly converted with ferocious power by Aubameyang.
Liverpool took a while to recover but they responded encouragingly with the Kop's new hero Origi having three chances, Roberto Firmino heading wide and Adam Lallana completely fluffing his shot in front of goal.
However, twice Aubameyang could have finished the tie before half-time as he missed Lukasz Piszczek's cross in the six-yard box and then scuffed wide a pass from Reus.
If Dortmund's start had been electrifying, Liverpool's opening to the second half was not far off matching it.
Within three minutes of the resumption, Origi poked Emre Can's through-ball past Roman Weidenfeller, and the newly-installed roof on the half-constructed new Main Stand nearly came off.
For a few fleeting moments there were dreams of an Istanbul-style comeback. Liverpool stunned AC Milan in the Champions League final 11 years ago, taking the trophy after being 3-0 down at one stage.
Dortmund are a team with far more energy than the Italians and they looked like fending off the resurgent Reds.
Within 10 minutes Reus had made it 3-1 to seemingly put the game beyond their hosts after another exquisite pass inside the full-back, this time by centre-back Mats Hummels.
The Anfield crowd appeared deflated and only regained some life with the appearance of Daniel Sturridge and Joe Allen on the hour.
The changes were just the lift the team needed, and when Coutinho drilled home from the edge of the area hopes of achieving a Mission Impossible rose again.
Sakho's stooping header 13 minutes from time made it even more likely as Dortmund tried to shut down the game, almost abandoning any attacking intent for the sake of self-preservation.
It proved in vain as in the first of four added minutes a clever free-kick saw Sturridge backheel the ball to Milner who crossed for Lovren to head in.
The famous old ground, which has seen many a glorious European night, shook to its century-old foundations.
Dortmund had one final chance to snatch it with the last kick of the match but substitute Ilkay Gundogan curled it just wide and once again there was an explosion of joy and relief.
At the final whistle Klopp lined up his players in front of the Kop to take the applause of a crowd who had more than played their part and the plaudits were well deserved both on and off the field.