The Dutch side trailed to Cristiano Ronaldo's fifth knockout-round goal in Juve colours with half an hour played, but they hit back through Donny van de Beek and Matthijs de Ligt to deservedly progress through.
Erik ten Hag's side have now seen off defending champions Real Madrid and many people's favourites Juventus to make it to the semis, where they will face the winners of the all-Premier League tie between Manchester City and Tottenham Hotspur.
Ajax, unbeaten in their last four away meetings with Italian opposition, looked a little nervy in the opening exchanges as Andre Onana's attempted clearance struck Ronaldo and flashed wide of the target.
The hosts struggled to find any sort of rhythm to their game in the opening quarter, though, and it took until the 22nd minute for Paulo Dybala to test Onana for the first time.
Van de Beek squandered a far better opportunity for Ajax, somehow failing to convert from close range when a loose ball bounced into his path, and his side were a goal behind soon after.
Ronaldo was left in space to power home a header from Miralem Pjanic's corner following some poor defending from the Dutch visitors.
Only six minutes were required for Ajax to respond, though, as Hakim Ziyech's shot deflected into the path of Van de Beek, who kept his composure to convert past Wojciech Szczesny - just his second goal in the competition.
On his 50th appearance in the Champions League, Szczesny produced a crucial save 10 minutes into the second half by getting a strong left hand to Ziyech's powerful drive.
That should have acted as a warning for the hosts, but that did not prove to be the case as Van de Beek - causing all sorts of problems in the final third - curled another shot goalwards a few minutes later that Szczesny was again equal to with his fingertip stop.
Ajax were enjoying a sustained spell of pressure and a third big chance came their way just after the hour when Dusan Tadic was picked out in the box.
Tadic, a key performer in Ajax's shock win at the Bernabeu in the last round, turned down the chance to shoot and instead saw his back-post pass cut out by Pjanic.
Juve were unable to cope with Ajax and the visitors were eventually rewarded with a third goal of the tie thanks to De Light's powerful header, outjumping Alex Sandro to get on the end of a corner.
That was Ajax's 155th goal this season in all competitions, a joint-club record, and it left Juventus needing two to progress.
David Neres miscued a close-range shot and Ziyech had a world-class goal ruled out for offside as Ajax tried to kill off the tie for good.
The near misses did not matter, though, as Juventus did not come close to scoring in the remainder of the match, ending Ronaldo's hopes of winning the trophy for a fourth season running.
JUVENTUS (4-3-3): Szczesny; De Sciglio (Cancelo 64'), Bonucci, Rugani, Sandro; Can, Pjanic, Matuidi; Bernardeschi (Bentancur 81'), Dybala (Kean 45'), Ronaldo
AJAX (4-2-3-1): Onana; Veltman, De Ligt, Blind, Mazraoui (Sinkgraven 11'; Magallan 82'); Schone, De Jong; Ziyech (Huntelaar 88'), Van de Beek, Neres; Tadic