Inter allowed Barca to have possession at San Siro and strike on the counter, which is what happened when Hakan Calhanoglu struck the winner for the hosts.
“I wouldn't say it was unexpected. I said yesterday it was a great opportunity against one of the best teams in the world," Inzaghi told Sky Italia.
“We played with aggression, determination, as it's the only way to beat these opponents. We'd been waiting a long time for this, I am happy for the fans and the club, but we've achieved nothing yet, it needs to be a wonderful night that marks the beginning of something.
“I was very, very disappointed with the result against Roma, but not the performance. Evenings like this one will do us good, we need to recover our energy now and we have ground to recover in Serie A.
“It can be the start, but we need to give a sense of consistency to this result and performance now."
Inzaghi was under pressure going into the game.
“I continue my work, there are moments that happen and I keep my head held high, because my staff, players and the club have done some great things over the last 15 months. I saw a squad that is solid and fights together, otherwise you don't get a result like this.
“I am happy, but these lads are the same ones who gave me two trophies just a few months ago and they know more than anyone we now have ground to make up."