Having won their opening two league matches, Inzaghi's side were level at the top with Juve going into the clash at San Siro but they fell to a 1-0 defeat.
Carlos Tevez's goal sent Milan to their first loss under the new head coach, who replaced Clarence Seedorf in May, but it could have been different had Keisuke Honda's first-half header gone in.
The Japan international met a Sulley Muntari cross perfectly at the back post but saw Gianluigi Buffon parry away his effort on goal.
That was Milan's only meaningful effort on goal in the match as Juve dominated proceedings and, although Inzaghi admitted his side struggled to cope with the defending champions at times, he was encouraged by their progression.
"I was angry because we really hoped that we'd be up to the level of an extraordinary team," he told Sky Sport Italia. "I told the players I am satisfied, as after two rounds we can't be on the same level as one of the strongest teams in Europe.
"If Honda's header had gone in, the game would've changed. We put two strikers on and tried to get back into it, but to no avail.
"Tonight should give us the awareness that taking it one game at a time we can grow and get closer to Juve and Roma, who have something more than the others.
"The goal was created by two great champions who decided a game that otherwise may well have ended 0-0 for a positive point. The defence had a great game tonight and closed off most of those runs.
"We are all disappointed not to get a result, but at the same time taking points off a side that won the Scudetto three years in a row and last season had 50 more points than us was asking a bit too much."