The visitors had lost on their previous two trips to Palermo but came into this meeting unbeaten in five and not having conceded a goal all season.
Their defence was breached in the opening three minutes, however, as Nemanja Vidic inexplicably lost possession in his own box, allowing Franco Vazquez to put the hosts in front.
Mateo Kovacic conjured a fine strike to bring Inter back into the game, despite their poor first-half performance, and they pushed for a winner in the second half but could not break through.
"Palermo were pressuring us and we held out, then for half an hour after the restart did everything right except the goal," Mazzarri told Sky Sport Italia.
"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.
"The Vidic error was like the ball going through a goalkeeper's legs – it happens once in 100 years. He is a very experienced player, but these things can happen to anyone.
"We pegged Palermo back for long periods and tried everything possible, but were unable to get the goal and could even have lost at the end.
"We are Inter, we have to go to every stadium aiming to win and with that attitude you can risk a defeat too."
Inter were heading for defeat as they struggled through the first half before Kovacic found the bottom corner from 20 yards.
The 20-year-old Croatian brought his team to life and is integral to so much of what Inter do in attack but his coach insists they cannot rely on him too heavily.
"I honestly did expect Kovacic to make this step up, as it was only a matter of time," added Mazzarri.
"We must all realise that the same players cannot play every three days for 90 minutes at full tempo. We have to all train well and keep mentally and physically alert to share out the resources."
