Jeremy Menez had given Milan the lead midway through the first half, but Joel Obi levelled for Inter just after the hour-mark.
Inzaghi said he expected a tough contest, particularly with the clash marking Roberto Mancini's first game back in charge of Inter.
But the 41-year-old felt his team had enough chances to claim all three points.
"It was an even match and we knew it would have been hard with the change of coach," Inzaghi told Milan Channel.
"I'm sorry for the fans, they helped us make the difference, we would have liked to have given them a win.
"[Goalkeeper] Diego Lopez was only made to work by [Mauro] Icardi by a misplaced pass by us and very little else.
"We conceded through a goal after two deflections and we finished the game with four successive corners and two chances through [Stephan] El Shaarawy and [Andrea] Poli.
"I'm disappointed but we have to move forwards. It was an excellent performance by the defenders, they were helped by the whole side."
Inzaghi, whose team are seventh, explained his decision to leave Giampaolo Pazzini on the bench.
He said he was about to introduce the 30-year-old, but felt El Shaarawy was just beginning to cause problems.
"I was about to bring Pazzini on, but then there was that chance by El Shaarawy," Inzaghi said.
"If I had taken him off he would have become even more discouraged, by scoring he would have been reinvigorated.
"But Pazzini has to stay calm, there's already the chance to play as of Sunday. I really liked this playing system a lot, when we have all the players available, we have a competitive squad."
