Nicolas Burdisso was sent-off after just eight minutes before Julio Cruz netted twice to give 10-man Inter a 2-0 advantage, pouncing on defensive errors.
"Inter deservedly scored twice. We seemed asleep and were not reacting at all," said Juve coach Claudio Ranieri. "We could not spread the play because Inter were waiting for us in the final 20 metres and remained very compact."
It all changed in the space of five minutes, as Alessandro Del Piero and substitute Jean-Alain Boumsong completed the comeback.
"This team is always present and the two late goals were very important. It was crucial to score away from home in this competition to make the second leg easier."
Ranieri also said: "It was a very strange game.
"We wanted to get a positive result and did start off well. With the extra man we tried to spread them and although we had most of the possession, it was sterile, as we never really tested Francesco Toldo."