The Giuseppe Meazza club edged ahead twice, but Palermo fought back as many times and two late goals from the Sicilians eventually proved to be too much for Inter.
"It's a pity that we lost, but we can improve," Alvarez said to Inter Channel. "We lost a match when we weren't playing badly. We're sorry about that but we showed our ability as a team to pick ourselves up and play well."
The Argentine admitted he was happy to have made his competitive debut for his new club.
He continued: "I'm pleased I have made my debut in one of the most important leagues, and for one of the most important clubs, in the world. Naturally the final score was disappointing.
"My position? I can play in many positions, I'll adapt to wherever the team want me to play."
Inter's next match is the Champions League game at home against Trabzonspor on Wednesday.