The hosts suffered an early setback when Alex was forced to leave the field through injury, with Filippo Inzaghi sending on Cristian Zapata in his place.
However, the Colombian had a match to forget, inadvertently putting Palermo in front when he headed beyond goalkeeper Diego Lopez from a 23rd-minute corner.
Things got even worse for Zapata when he was easily brushed aside by Paula Dybala three minutes later as the Argentine stole in to lift the ball over Lopez and double the visitors' advantage.
Misfiring Fernando Torres failed to enhance his stuttering reputation, lacking conviction when opportunities fell his way, while the introduction of Stephan El Shaarawy for the second half failed to have the desired effect.
The defeat is just Milan's second since Inzaghi took charge, while for Palermo the result represented their first away win since promotion to the Italian top flight.
Giovanni Marchese's sublime 20-yard half-volley in the 21st minute pulled Genoa level, before Iago Falque turned on the spot in the six-yard box to put the visitors in front two minutes later.
Silvan Widmer made sure it was level at the break by sending a side-footed effort beyond goalkeeper Mattia Perin after 41 minutes but Alessandro Matri's low drive nine minutes into the second half restored Genoa's lead.
Elsewhere, Atalanta's trip to Torino and Sassuolo's match at Chievo both ended in goalless draws.
