Central defender De Ligt, the Ajax captain at just 19 years of age, headed the winner in a 2-1 victory in Turin – the Amsterdam club winning 3-2 on aggregate.
Ajax are the first team in Champions League history to make the semi-finals after surviving three qualifying rounds, and will now meet Manchester City or Tottenham in the last four.
“I can’t describe what kind of emotion we have at the moment,” De Ligt told BT Sport 3 after Ajax had added Cristiano Ronaldo’s Juventus to their list of Champions League victims, having previously eliminated holders Real Madrid.
“It’s unbelievable what we achieved, we are a squad from the Netherlands and we show the world what we can do.
“We showed it in the Bernabeu (against Madrid), we showed it against Bayern Munich (at the group stage) and now we showed it against Juventus.
“But we are never satisfied and the next game is also going to be important, we’ll see what happens.
“There is so much potential in our team and we are all still young. You see us grow every game.”
Ronaldo headed Juventus into a 28th-minute lead with his 126th goal in the competition and his ninth in seven meetings against Ajax.
But Donny Van De Beek levelled six minutes later and De Ligt’s winner midway through the second half rewarded Ajax’s superiority.
“It was a great corner kick,” De Ligt said.
“I jumped and I know I can jump high and head the ball well. That was the perfect moment to score.
“We had so many chances to score at the end, it is 2-1 but it could be 3-1 or 4-1.
“The only thing we lack is that we have to score more, but that is a big compliment to us as a team.”
Ajax will definitely meet English opposition in their first Champions League semi-final since the 1996-97 season and the days of Edwin van der Sar, Jari Litmanen and Marc Overmars.
But asked whether he wanted to play City or Spurs, De Ligt said: “At the moment I don’t have any preference.
“We will see who is coming to the semi-final and then we have to show the same.”
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Lionel Messi’s first-half double – the first a trademark finish, the second courtesy of a blunder from David De Gea – plus Philippe Coutinho’s 61st-minute wonder goal gave a stark reminder of the difference between the two sides.
Solskjaer will now turn his attention to the domestic front, where United will vie for a top-four spot in the Premier League, but the Norwegian knows he faces a major rebuilding job this summer and beyond.
He said: “We did well to get here and you could see the difference between the two teams.
“We know there’s work to be done. I’ve said all along this isn’t going to change overnight and the next few years are going to be massive for us to get to the level where Barcelona and other teams are at the moment.
“We have to look at ourselves, we know we have a job to do and we are getting on with it.
“We need to get the best out of each other, create an environment of top, top, top-class attitude, world-class attitude every single day. Because we’ve got good players and I’ve got good players to work with.
“We’ve done really, really well to get to the quarters and challenge for the top four.
“So we’ve got a rebuilding job but it starts with the coaches and players, and of course one or two additions to the squad will happen in the summer.”
Following last week’s 1-0 defeat at Old Trafford, United knew they needed to score against a side who, coming into Tuesday night, had not lost in their last 30 matches at home in the Champions League.
Solskjaer felt United needed to display a fearless and attacking mindset in the intimidating atmosphere, similar to the one that saw them overturn a two-goal deficit at Paris St Germain in the last 16 last month.
They made an enterprising start as Marcus Rashford struck the top of the crossbar in the first few seconds but were thoroughly second best after five-time Ballon d’Or winner Messi opened the scoring after 16 minutes.
Dispossessing Ashley Young on the flank, Messi cut inside and curled beyond De Gea, who four minutes later allowed the Barcelona captain’s tame right-foot effort to slide under him.
That effectively ended the contest – while Coutinho added gloss to the scoreline with a superb strike from distance after the interval – though Solskjaer was understandably reticent to criticise a goalkeeper who has so often come to United’s aid in recent years.
Solskjaer said: “That’s unfortunate with goalkeepers. When you make an error, it can get highlighted and that’ll be remembered.
“But he made some fantastic saves as well, which all in all, his contribution to the team is good but sometimes that happens in football.
“The quality of their finishing was absolutely outstanding. We started well in the first 15 minutes, we felt we’ve got something here and then in four minutes they scored two goals.
“The attitude was right, we were fighting against a good team and good players out there.
“Messi is an exceptional talent, him and Cristiano (Ronaldo) have been the best players in the world in the last decade.
“He showed why we all think that and why he’s won so many titles and why this team have won so many Champions Leagues (four since 2006).
“You can prepare as much as you like, if you give him space and time around the goal he’ll create and score goals, he’s a fantastic player.”
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Messi got the breakthrough in the 16th minute to put his side firmly in the driving seat of the quarter-final second leg before a howler from David De Gea handed him a second four minutes later.
Philippe Coutinho’s effort from range on the hour mark sealed progress to the semi-finals for the Catalans with a 4-0 aggregate score.
Here, we analyse Ernesto Valverde’s tactical input.
Goals – 3
Shots – 13
Shots on target – 6
Goals conceded – 0
Possession – 66%
Dribbles – 14
Tackles – 16
Lionel Messi has scored more Champions League goals against English clubs than any other player in #UCL history:— Squawka Football (@Squawka) April 16, 2019
9 vs. Arsenal
6 vs. Man City
4 vs. Man Utd
3 vs. Chelsea
2 vs. Spurs
Just waiting to add Liverpool now... pic.twitter.com/8EqgWgb1Hk
TACTICAL TALKING POINT
With Messi in the form that he was in on the night, it’s only natural that he would take all the plaudits. Whatever spotlight remains would undoubtedly be fixated on United’s glaring individual errors and general lack of quality. Spare a thought for Sergio Busquets though who almost single-handedly got Barca to settle down after a shaky start.
The experienced holding midfielder was made to sit a lot deeper after United’s early pressing and threat in the transition. He did that expertly, slotting in between the two centre-backs and helping Barca play their way out of danger. As the game wore on, that became a position of strength for the hosts. The Spaniard was critical to their build-up play, often leaving United’s forwards chasing shadows.
Targeting United’s right side
As Barcelona increasingly gained control of the game, they started to target the right side of United’s defence, a ploy that was particularly noticeable after half-time. Especially after winning possession, their immediate passes were directed at finding either Coutinho or Jordi Alba. The duo threatened on a couple of occasions before linking up for Barca’s third.
Valverde did what has come to be expected of him. His 4-4-2 system is tailor-made for Messi to run the show and run the show he did. The tactician must be credited for the subtle tweaks he is known to make during games which help his side along their way. Has to be said that he benefited hugely from United’s own errors though, ultimately making things rather simple for him.