Juve, who have yet to concede a goal in Italy's Serie A this season after six games, had to fight back twice from a goal down before securing the win with a late goal from Francisco Conceicao.
"I'm really happy, especially for the team," said Juve forward Dusan Vlahovic, who twice equalised for the Italian side.
"We are becoming a real unit, we fought together, kept pressing even when we were down to 10 men, and in the end, we deserved to win.
"It was a great match, and fortunately, we came out on top."
The Italians, who beat PSV Eindhoven in the opening match of the competition, suffered a disastrous start with a double injury blow when captain Gleison Bremer was substituted following a challenge with Lois Openda in the sixth minute and Nicolas Gonzalez was also forced off.
They were down to 10 men when keeper Di Gregorio was dismissed with a straight red card on the hour for a handball outside the box.
"It was a good match, even in the first half where we made some mistakes in the details," said Juve coach Thiago Motta. "In the second half, we did better, even playing with 10 men. It was a great performance and a great win.
"Courage from the boys, they were determined to push forward and hurt the opponents. Even with a man down, we had that feeling, so why step back? We kept pushing, and we did it well right until the last minute."
