Trailing by three goals from the first leg, no one gave the Reds much hope, but two goals apiece from Divock Origi and game-changing half-time substitute Georginio Wijnaldum saw them progress 4-3 on aggregate.
In a truly remarkable and intense game not only did Liverpool erase that disadvantage inside 56 minutes, they went on to score a winner and, remarkably, kept Lionel Messi quiet.
Liverpool captain Jordan Henderson admitted the players had expected a difficult game with most people having written them off.
He said: “I don’t think many people gave us a chance. We knew that it would be difficult but it was still possible for us.
“The belief we’ve got in the changing room. At Anfield we knew we could put on something special. We wanted to start fast, we got an early goal, which helped. We knew that if we gave everything, showed a bit of personality, a bit of heart (we could win).”
Manager Klopp said: “The whole game, the whole performance was too much. It was overwhelming. I watched in my life so many football games, but I can’t remember many like this. Winning is already difficult, but winning with a clean sheet I don’t know how the boys did it.
“I saw James Milner crying after the game on the pitch, it means so much to all of us.
“There are more important things in the world, but creating this kind of atmosphere is so special.”
For only the fourth time in the competition’s history a team was eliminated from a knockout tie having won the first leg by three or more goals, the last occasion being Barca against Roma in last season’s quarter-final.