Ronaldo was rested before Juve's second-leg last-16 tie and he subsequently scored a hat-trick against Atletico Madrid to overturn a first-leg deficit.
But after defender Ashley Young let Messi in for an opening goal in the 16th minute at the Camp Nou, the pressure ratcheted up, with no more room for error.
Such a commanding performance from 19-year-old De Ligt - he made a game-high 10 clearances - has become commonplace despite his tender age.
A first half goal from Juventus' star man Cristiano Ronaldo, whose €100m purchase over the summer saw fans buy more than €50m worth of shirts in the subsequent 24 hours, was not enough to win after Ajax hit back with two strikes of their own.
"The image (we gave) was spectacular. He's disappointed like all of us", said Allegri.
"I still need to decide who starts in attack, it could be Dybala or Kean or neither of them". The important thing is that we achieved our goal. Last week, we controlled the game, but were unable to score more than one goal.
After not managing a single shot on target at Old Trafford in the first leg, United looked a lot more unsafe going forward at the start.
"Now, two years later, he's a really fantastic player", Mourinho said.
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Ronaldo scored a hat trick in the previous round against Atletico Madrid to nearly single-handedly overturn a two-goal deficit, and had also found the back of the net in Amsterdam.
The pressure finally told after 67 minutes as De Ligt towered above Alex Sandro and Daniele Rugani to head Lasse Schone's corner past a helpless Szczesny.
Ajax humiliated Madrid 4-1 in the Spanish capital en route to the quarters and the in-form Dutch side also claimed the scalp of Juve thanks to Tuesday's 2-1 second-leg victory in Turin. Liverpool has a pair of Dutchmen in Virgil van Dijk and Georginio Wijnaldum, two players who know a lot of the Ajax players well but never played for the club.
It's the first time the Champions League semi-final will not feature the Portuguese superstar for nine years, according to the statistics website Opta.
He has won the title five times and scored a 97th-minute penalty for Real Madrid against Juventus last season in the quarter-finals. It seems he will be on the market and when we talk to the coaches we will see if we're in for him or not.
That means Ajax will now have a full week of rest between their home clash with Vitesse on April 23 and their visit to Spurs, with Tottenham facing West Ham in the Premier League just three days before the Champions League game.
Liverpool have won three of their four penalty shoot-outs in Europe while in all competitions they have won 14 and lost four.