Nicolas Lopez emerged from the bench to fire a low leveller two minutes from time to deny the hosts a victory which had seemingly been sealed by Mauro Icardi's double.
Mazzarri's side looked to be on the way to victory despite Luca Toni's early opener, as Icardi volleyed a first-half equaliser.
There was no reprieve though as Lopez made it three matches without a win for Mazzarri, who has been under pressure with media reports suggesting former defender Sinisa Mihajlovic could be brought back to San Siro as coach.
Verona made a much stronger start and went ahead with a slick counter-attack - led by the returning Emil Hallfredsson - that saw Lazaros Christodoulopoulos nod into the path of Toni and the veteran striker expertly volley home from six yards.
It lifted Inter and they almost completed the turnaround moments later as Zdravko Kuzmanovic's 25-yard curling shot cannoned off the post.
The end-to-end nature of the game continued with a superb reaction save from Rafael denying Nemanja Vidic's powerful header before an inswinging Christodoulopoulos cross evaded everyone but not the razor-sharp Handanovic in Inter's goal.
It took less than three minutes from the restart for Inter to take the lead as Palacio found space and drilled to the far post where Icardi tapped in his seventh league goal.
Verona had a chance to draw level from the spot after 50 minutes when Medel handled Juanito's shot, earning him a second booking.
However, Handanovic guessed right and saved Toni's penalty to register his third spot kick save of the season.
