Mazzarri's team then saw two goals controversially disallowed, but he has acknowledged that Palermo also had plenty of chances to claim the three points.
"We were beating a record of clean sheets, but managed to gift a goal within three minutes, so nobody needs to talk about that anymore," he told Sky Sport Italia.
"Palermo were pressuring us and we held out, then for half an hour after the restart did everything right except the goal. We used up a lot of energy to go for a win and I moved to a four-man defence, but Palermo were fresher and we risked defeat at the end.
"It was a tight game and one incident could send the result one way or the other. I know what I need to tell the lads and where we can improve, but it was tough and I accept the result."
Inter face Atalanta in their next Serie A match on Wednesday evening.