Liverpool 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.