Dejan Lovren was the stoppage-time hero as Liverpool scored three times in the final 24 minutes in a magical Anfield comeback to dump Borussia Dortmund out of the Europa League.
After a 1-1 draw in the first leg of the quarter-final tie, Dortmund seized control with goals from Henrikh Mkhitaryan and Pierre-Emerick Aubameyang inside the opening nine minutes on Thursday.
Divock Origi got a goal back early in the second half, but Marco Reus' ruthless finish appeared to have killed off the tie.
However, Philippe Coutinho's impressive strike and Mamadou Sakho's header set up a frantic finish at a raucous Anfield, with Dortmund clinging on in an attempt to progress on away goals.
It looked like they would do just that until Lovren headed home a winner in the 91st minute, prompting delirious celebrations from Liverpool's players and manager Jurgen Klopp, who haunted his former club.
Klopp's men move into the semi-finals after a night that will live long in the memory, alongside historic European comebacks against the likes of Saint-Etienne, Olympiacos and AC Milan.
A stunned Dortmund side – who sit second in the Bundesliga – will likely have to win the DFB-Pokal to avoid finishing the campaign trophyless.
A quick break saw Shinji Kagawa – one of the visitors' two changes from the first leg – and Gonzalo Castro combine to set up Aubameyang, whose volley from six yards was pushed away by Simon Mignolet, allowing the onrushing Mkhitaryan to side-foot home the rebound.
Origi – selected ahead of Daniel Sturridge in attack as at Signal Iduna Park – had a shot deflected wide and Adam Lallana miskicked badly from a good opening as the hosts attempted to reply.
Origi flicked another effort wide and Roberto Firmino – Liverpool's only change from the tie in Germany - was off-target with a diving header.
At the other end, Aubameyang could not quite get enough on a sliding attempt and Mignolet got down sharply to keep out Kagawa as Klopp's men, despite having made a spirited response, went into half-time with it all to do, having failed to test Dortmund keeper Roman Weidenfeller.
