The visitors set about trying to pull back the deficit, Yuto Nagatomo forcing a corner in the 10th minute before taking a shot from outside the box following the set piece, but was unable to hit the target.
Achraf Lazaar's cross then caused havoc in the Inter box nine minutes later, Walter Mazzarri's side managing to scramble the ball away.
The Nerazzurri struggled to find their way into the game for most of the first half, though Fredy Guarin was unlucky not to hit the target with a long-range strike just after the half-hour mark.
Palermo kept plugging away against their shellshocked opponents, but were unable to add to their early goal.
They would soon regret their wastefulness in the final third, as the Nerazzurri equalised with just three minutes to go, courtesy of Mateo Kovacic's opportunist strike.
Palermo then had another scare when Vidic headed the ball past Stefano Sorrentino and into the net moments later, but the referee disallowed the goal for offside.