The five-time European champion beat Lionel Messi's side 4-0 on Tuesday, overturning a 3-0 first-leg defeat to progress to the final for the second consecutive year, AFP reported.
Stand-in striker Divock Origi and substitute Georginio Wijnaldum both scored twice as Klopp was forced to cope without injured star attacking duo Mohamed Salah and Roberto Firmino.
"Do not adjust your reality. This really is happening. There have been glorious, entirely improbable games in Liverpool's European history, including the mind-bending highs of Istanbul," said the Guardian.
"But this was something else, an effort of will that, frankly, took the breath away. On a rapturous night Liverpool's season of chasing to the end narrowed first to a fine point, then burst into the most extravagant life as a 1-0 half-time lead against Barcelona became two, then three, then four.
"With 79 minutes gone, the most celebrated team of the modern age had been reduced to bunch of mooching, stumbling yellow-shirted spectators."
Liverpool has gone toe-to-toe with Manchester City all season in one of the most gripping Premier League title races ever but look like falling just short of winning its first English top-flight title since 1990.
Now theReds have the chance to overshadow City's achievement and be crowned kings of Europe for a sixth time in Madrid next month.
For the Daily Mail's Martin Samuel, it was better even that Liverpool's comeback in Istanbul from 3-0 down against AC Milan to win the Champions League final in 2005.
"At the end of this wonderful, unbelievable, fantastical game, Jurgen Klopp linked arms with his players, facing The Kop as the whole of Anfield, including some among the bereft Catalan enclave, sung 'You'll Never Walk Alone'," he said.
"One had the feeling this was the moment he had been working towards since the day he set foot on Merseyside. This spirit. This togetherness. This performance. This passion, this emotion: It was all here, every last drop of what he wanted to achieve. And yet, there is still such a long way to go."
The Telegraph said Liverpool's success could be written in the stars.
"No team does this in the second leg of a Champions League semifinal without wondering if the hand of destiny is not ushering them down an alternative path to glory to the one they have chased all season.
"On Monday night, Manchester City edged the Premier League just a little further from Liverpool's grasp which felt monumental – and then came Tuesday night when the response at Anfield was, quite frankly, stupendous."
And the Independent said Liverpool had produced a night of pure fantasy.
"When the curtain comes down on this season, and for many years to come, Liverpool fans will bond over the belief and the disbelief of this night.
"'The Barcelona game', they'll say, and that'll be all they need to say. A night when Anfield heaved with the heft of the impossible, when a crowd of thousands and an audience of millions lost itself in the mad, dangerous intoxication of football."
'Biggest embarrassment of all time'
Meanwhile, The Spanish and Catalan press were in unforgiving mood after Barcelona’s 4-0 surrender to Liverpool.
“The biggest embarrassment of all time,” said the front cover of Barcelona-based newspaper Sport on top of a black canvas after the 4-3 aggregate defeat, adding “Barca writes the darkest page in its history”, Reuters reported.
“A historic failure”, said Marca, listing all the factors in Barca’s favor before the game which deepened the scale of its defeat and denied Messi the chance to lift the trophy he had clearly targeted at the start of the season.
“They were 3-0 up, Liverpool’s stars were injured, the league was won, the final is in Madrid, Real Madrid was already out and they had the best player in the world.”
Parallels were drawn with Barca’s eerily similar capitulation in last season’s competition to AS Roma in the quarterfinals, when the Catalans crashed out on away goals following a 3-0 defeat in Italy after leading the tie 4-1.
Catalan newspaper Mundo Deportivo also talked of an embarrassment and slammed Ernesto Valverde’s side for switching off for the fourth goal, scored by Origi but set up by the quick thinking of Trent Alexander-Arnold.
“A ridiculous goal conceded from a corner will be the epitaph of this team, which repeated the mistakes of Rome,” said the newspaper’s cover.
Barca’s Luis Suarez and Sergio Busquets apologized to fans after the game, as did Valverde, and there is a feeling that the coach is unlikely to survive this latest European failure despite overseeing the team’s domestic dominance.
“The Champions League is too big for Valverde,” said an editorial in Sport.
“Without doubt Valverde holds a lot of responsibility for this disaster, because when the moment of truth arrived he made the same mistakes.”
The players did not escape blame either.
Marca’s ratings, which judge individual performances by awarding players between one and three stars, gave seven of Barca’s starting 11 no stars at all.
Sport’s player ratings meanwhile gave no player more than three marks out of 10, while midfielders Ivan Rakitic and Philippe Coutinho were each given a rating of zero.