Barcelona had their opportunities on an unforgettable occasion but at times, the La Liga champions looked powerless as a rampant Liverpool wrote another page in their illustrious history to set up a showdown with either Ajax or Tottenham Hotspur.
Liverpool set the tone for the task ahead by almost going ahead after 51 seconds, with Jordan Henderson being denied by a last-gasp interception at the back post, but it did not take long for the home side to get themselves back in the tie.
Liverpool remained on the offensive although it soon became apparent that a gung-ho approach would quickly be undone, and Lionel Messi saw a shot tipped over the crossbar by Alisson after meeting Alba's cutback.
Messi quickly turned provider for Philippe Coutinho to have a low curling effort parried away by Alisson, while last week's two-goal hero dragged a half-volley wide of the goalkeeper's post from the edge of the area.
Robertson had a strike from an acute angle kept out by Ter Stegen at the other end, but Liverpool had quickly become second best to a side intent on netting what would have probably been a tie-clinching goal.
Liverpool rallied as the first half moved closer towards a conclusion, but Klopp's side were fortunate to make the break without conceding an equaliser during four minutes of added-on time.
A mistake from Xherdan Shaqiri eventually resulted in Messi dragging a shot just wide from distance, while Alba got on the end of a pass from the Argentine only for Alisson to make a timely block.
Liverpool were forced into a change before the restart, with Robertson failing to shake off a knock sustained after a kick from Luis Suarez, but Klopp's men again pushed forward for an early goal with Virgil van Dijk being denied by Ter Stegen after a clever back-heeled volley.
Suarez saw a shot from 14 yards parried away by Alisson as Barcelona again showed intent in the final third, but Liverpool soon pulled a second goal back thanks to Wijnaldum.
Trent Alexander-Arnold's low cross from the right was deflected into the path of the substitute, who forced a first-time effort under Ter Stegen from by the penalty spot.
The home supporters were beginning to sense something special was taking place in front of them, and the comeback was soon complete when Shaqiri crossed for the Dutchman to power a header into the top corner from eight yards.
The capacity crowd inside Anfield were in delirium as Barcelona struggled to comprehend what had just taken place, but Ernesto Valverde's side soon gathered themselves to take control of the game ahead of the final quarter.
Messi was always a threat - regardless of whether he was in possession - and the forward almost netted his third of the tie, with Alisson having to make a smart save at his near post.
However, Liverpool remained ambitious when on the ball, and the quick-thinking of Alexander-Arnold led to Liverpool remarkably taking the lead in the tie with 11 minutes remaining.
Barcelona were attempting to organise themselves in preparation for a corner, but Alexander-Arnold sent a low pass across the box for Origi to find the top corner as the visitors desperately appealed for it to be retaken.
Liverpool had to withstand constant pressure from Barcelona during a nervous end to an enthralling contest, but the visitors failed to create the openings which had come so easily during the first hour.
The Premier League side spent the closing 90 seconds in the far corner of the pitch, and the full-time whistle was greeted with a crescendo of noise which will surely only be eclipsed if Liverpool win the league title on Sunday afternoon.
LIVERPOOL (4-3-3): Alisson; Alexander-Arnold, Matip, Van Dijk, Robertson (Wijnaldum 46'); Henderson, Fabinho, Milner; Shaqiri (Sturridge 90'), Origi (Gomez 85'), Mane
BARCELONA (4-3-3): Ter Stegen; Roberto, Pique, Lenglet, Alba; Vidal (Arthur 75'), Busquets, Rakitic (Malcom 80'); Coutinho (Semedo 60'), Suarez, Messibody check tags ::