Del Piero hit both of Juve's goals in the Champions League shock, with his second coming via a free-kick which left Casillas flat-footed.
The Spain No1 said: "The second goal was my fault and I've been unfortunate in that action.
"But we had several chances, for example (Mahamadou) Diarra had two very clear ones. They, with very few chances to score, have earned a vital three points.
"Right now we have to play in Belarus (against BATE Borisov) and secure second place."
Meanwhile, Real midfielder Guti added: "It is a tough result for us. They played their game and we failed to unlock them.
"There is nothing more to say. It is important to run, but also we have to play a quality game."