After Barcelona won the first leg 3-0 at the Nou Camp, few gave Liverpool a chance of scoring the goals they needed to progress, let alone preventing the La Liga champions from scoring themselves.
But an early strike from Divock Origi, who tapped in after Marc-Andre ter Stegen failed to hold Jordan Henderson's shot, gave the vociferous Anfield crowd the belief that another European fightback from the Reds could be on the cards.
Georginio Wijnaldum, on as a substitute for the injured Andy Robertson, made it 2-0 with a low drive nine minutes after the break before the midfielder almost immediately made it 3-0 on the night and 3-3 on aggregate with a superb header.
And Origi, starting the game because of injuries to Mohamed Salah and Roberto Firmino, completed a remarkable victory for Liverpool when he swept Trent Alexander-Arnold's low corner into the back of the net late on.
The win means Liverpool are into the Champions League final for a second successive year and they will play either Ajax or Tottenham in Madrid on 1 June.