Gala won the first-leg of their Champions League playoff 5-2 in Istanbul on Tuesday night.
Del Piero said on Sky Italia: "Luciano Spalletti talked about everything, from the players' awareness onwards, outlining one concept: if Juventus don't play at their best, they risk making trivial errors, which then punish you.
"But also of lacking presence on the pitch and personality; in the end, the others play football and overwhelm you.
"Galatasaray played an excellent game tonight. They had (Leroy) Sané and (Mauro) Icardi on the bench, to name a few, but whoever played had an exceptional performance.
"They played better than Juve. (Luciano) Spalletti admitted it, Juve really struggled today: perhaps mentally and physically they're starting to pay the price for what happened on Saturday, in the sense that it was a tough match.
"When you play games every three days where you have to be at your best and do well, that's when you see a team's true potential . If Juve can't do this well, they'll pay the price."