The young Dutch side backed up their 4-1 humbling of Real Madrid at the Bernabeu in the last 16 by stunning Cristiano Ronaldo and Juve 2-1 in Turin.
The four-time winners will face either Manchester City or Tottenham in their first last-four appearance since 1997.
“It’s bizarre, it’s not normal. I have no words for it,” the 19-year-old De Ligt told BT Sport.
“We have shown once again that we are very strong and that we can make it difficult for large teams.
“We have beaten Real and Juventus, two intended title candidates… I wasn’t born the last time we reached the semis.”
Ronaldo had put Juve ahead before the half-hour mark, but Donny van de Beek quickly levelled, and De Ligt finished the job in the 67th minute as Ajax utterly dominated the second half.
“I was going a bit crazy because we couldn’t convert any of the chances,” De Ligt told Dutch television channel Veronica TV.
Barcelona-bound Frenkie de Jong turned in another strong display in midfield, with Dusan Tadic and Hakim Ziyech continuing their excellent form up front.
“In the first half they were slightly better and deserved to take the lead,” said De Jong.
“But apart from the first five minutes of the second half, we dominated after the break and were well worth our victory.
“In fact, we probably should have won by more.”
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