The Nerazzurri boss has come under fire after his side's inconsistent start to the season, with the 2-1 loss to Roma leaving it with 12 points from eight Serie A games. Inter has conceded 13 goals in those eight games too.
Defeat on Wednesday (AEDT) would have left Inter perilously placed in third in its Champions League group, having lost to Bayern Munich 2-0 at home on the opening matchday.
"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," Inzaghi said. "I wouldn't say it was unexpected. I said yesterday it was a great opportunity against one of the best teams in the world.
"We played with aggression, determination, as it’s the only way to beat these opponents."
Inzaghi, who took over after Antonio Conte's split with the club after its 2020-2021 Serie A title, lifted two trophies last season, the Coppa Italia and Supercoppa Italiana, but ultimately fell short in the race for the Scudetto to Milan.
"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," he said.
"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."
Inter forward Lautaro Martinez promised the result would usher in charge.
"Tonight is a special evening because it was a victory we needed," the Argentine star said. "Because of the way it arrived we enjoy it even more.
"We were in a period when the results were not coming, today we needed a win and it has arrived. From here on, things will change."
Goal-scorer Hakan Calhanoglu, who netted his first Champions League goal since September 2016, added: "This game can really shake us up and change our season, because we want to build on it.
"As for qualification, we’ve got to stay calm and take it one game at a time."