Walter Mazzarri's men came into the game in fine form, unbeaten in their first five competitive fixtures without conceding a goal and having thrashed Sassuolo 7-0 last weekend.
But they fell behind to Franco Vazquez's early strike and were second best in the first half before Mateo Kovacic came up with something from nothing to get them back into it before half-time.
The Milanese outfit twice thought they had taken the lead, either side of the break, but were denied on both occasions by an offside flag, while the hosts were also frustrated by a disallowed goal in the dying embers of the match - a decision that coach Giuseppe Iachini protested so fiercely that he was banished from the touchline.
The result leaves Palermo looking for their first league victory since their return to the top flight and Inter with five points from three games.
Inter's confidence was clear from the first whistle as Danilo D'Ambrosio, who scored the winner against Dnipro in the UEFA Europa League on Thursday, drove down the right from the kick-off and won a dangerous free-kick.
But, with that set-piece failing to yield an opening, they were behind just a minute later. Nemanja Vidic hesitated with the ball at his feet in the box, when a routine clearance was all that was required, and Vazquez picked his pocket before lifting over Samir Handanovic from a tight angle to give Palermo a shock early lead.
That setback sapped the self-assurance from Inter's play and they struggled to find their rhythm, uncomfortable in possession all over the pitch.
Mazzarri's charges played themselves into trouble time and again, and would have been two down had Michel Morganella made any contact with Achraf Lazaar's teasing left-wing cross in the 19th minute.
Palermo continued to dominate the midfield, stifling Inter until the 42nd minute when Kovacic reacted first as the ball broke from a tackle to sweep it low into the corner from 25 yards to level the match against the run of play.
Vidic then thought he had atoned for his error and completed an unlikely turnaround two minutes later but his header from a Fredy Guarin free-kick was ruled out, with Pablo Daniel Osvaldo in an offside position.
The visitors had another goal chalked off for offside in the 51st minute, Vidic and Guarin combining again from a free-kick before Mauro Icardi, who appeared to be level with the last defender, converted.
Inter were in the ascendency but the final ball continued to escape them and it was Vazquez who came closest to a deciding goal, hitting the crossbar with a deflected shot in the 72nd minute before firing wide a minute later.
It was Palermo's turn to feel aggrieved in the 88th minute when substitute Andrea Belotti beat Handanovic on the break, only to have been judged to have fouled Yuto Nagatomo in the build-up.
Iachini was so incensed by the decision that he was sent to the stands for his protests but his side earned a well-deserved point.
