Hernanes had an opportunity for Lazio in the first minute, but couldn't find the target to give the away side the lead.
Palermo drove forward with real purpose for the first time after 20 minutes, but Fabrizio Miccoli could only hit the side-netting with his shot.
Ledesma had a chance to double his side's lead, but his effort was well saved by the goalkeeper.
Arevalo Rios broke free last into the second half for the home side, but his strike went wide of the post.
Palermo found an equaliser after 70 minutes when Andrea Dossena set up Rios, who finished well to bring the scores level.
The home side then completed an astonishing two-minute turnaround when Paulo Dybala made it 2-1 after being set up by Miccoli on 71 minutes.
Lazio began to push forward in search of an equaliser, with Floccari and Ledesma seeing their efforts go wide of the mark.
The referee then awarded Lazio a penalty on 84 minutes, which Hernanes slotted home to earn a point for his side.