Cristiano Ronaldo gave his side a further edge when he opened the scoring with a header in the 28th minute on Tuesday.
But from there, things went south for the Italian champions, who conceded an equaliser to Donny van de Beek just eight minutes later.
With the tie level, Ajax showed more of the initiative and were rewarded with a second goal on 67 minutes, with Matthijs de Ligt grabbing a vital header for the visitors.
The Dutch side would hold on from there, booking a place in the semi-final with a surprise 3-2 aggregate win.
Pirlo, who starred for Juve between 2011 and 2015, felt his former side looked tired and timid, while calling out two of their top performers.
“Juve did well for the first half-hour, when pushing them high and forcing Ajax to play the long ball," Pirlo told Sky Italia.
“After the equaliser, I saw Juve too timid and timorous, especially in players who ought to have character like [Miralem] Pjanic and [Federico] Bernardeschi, who looked tired."