Inzaghi: Beating Real puts Milan among world's best
Inzaghi's men sauntered to the victory courtesy of a Stephan El Shaarawy brace.
Milan are seventh in Serie A, trying to improve on their eighth-place finish in 2013-14 - their worst standing since 1998.
And Inzaghi said their recent results, which included a win over Napoli, a draw at Roma and their victory over European champions Real, held them in good stead to climb the Serie A table.
"We knew that if we didn't have the right attitude in this game we could have conceded up to five goals and that would have been a shame considering how far we have come after the games against Napoli and Roma," Inzaghi said post-match.
"We played three games against Napoli, Roma and Real Madrid and we come out with two wins and a draw - this means that we can play on a par with everybody.
"Although, yes, Real Madrid are in a league of their own... but the spirit and the effort we showed was great.
"These guys had trained for only three days after a week break and it didn't look like that."
Inzaghi was quick to redirect his side's focus back to their return to domestic duties, when they host Sassuolo on January 6.
"If we lose against Sassuolo we are going to forget about this game very soon, I can tell you," the former Juve and Milan striker said.
"In fact, we have to forget about this game tomorrow straight away.
"Yes, we must remember this as a nice evening and we must carry with us the fact that it was good to face players who can win the Ballon d'Or every year.
"So winning this game means that we get a big boost of confidence - it's a sign that we are growing and that's the best thing of tonight."
El Shaarawy declared his performance a "good achievement", but was also eyeing off continuing Milan's good form on their return to Italy.
The Italy international also dismissed speculation he was keen to depart San Siro.
"As I said before, I want to stay here at Milan," the 22-year-old said.
"I want to try and do well starting from next game onwards.
"And then I want to achieve a third place finish with this team."