Cristian Zapata inadvertently opened the scoring for the visitors by heading into his own net from a corner after 23 minutes, before Paulo Dybala doubled the advantage three minutes later.
Milan lacked quality in the final third and Inzaghi conceded his side were not at their best.
"This is football," he is quoted as telling Sky Sport Italia. "I said yesterday that Palermo were tough, having done very well at Napoli [in a 3-3 draw].
"We hadn't started that badly with [Andrea] Poli and [Fernando] Torres close, so if those chances had gone in it would've been a different game, but Palermo broke the deadlock and deserved the victory. We tip our hat to them.
"Nights like this can also help you grow. I take responsibility and from tomorrow we must roll our sleeves up.
"We can't wait to get back on the field against Sampdoria [next Saturday] to make up for this performance.”
Zapata had come on in place of the injured Alex after three minutes, but endured a match to forget as he scored the own goal and was found wanting for Dybala's second.
"Alex went off straight away, Zapata had to go on without a warm-up and he immediately felt some pain too," added Inzaghi. "It was just one of those nights.
"I am the coach, I'll try to work out where I went wrong, but this squad must be allowed to grow in good time. It was a heavy blow, but we'll get back up."
