Pinilla headed over with the goal at his mercy from Luca Antonelli's unselfish cross after 76 minutes, although Gasperini can at least take heart from the fact his side extended their unbeaten run to seven games in all competitions having gone behind to Paulo Dybala's fine early opener.
Andrea Bertolacci restored parity and Franco Vazquez was denied by the frame of the goal in an entertaining clash before Pinilla endured his moment to forget.
Palermo took the lead after just seven minutes as Dybala cut in from the right flank, stole a yard on his marker and looped a curled effort across goalkeeper Mattia Perin to net for the third game in a row.
Bertolacci hauled the sides level on the half-hour mark when he dispossessed Enzo Maresca and dispatched a low, right-footed shot from just outside the penalty area that nestled in the bottom corner.
Antonelli was denied by Stefano Sorrentino, while at the other end Perin's superb one-handed save stopped Dybala from edging Palermo back in front five minutes later when the striker was through on goal.
The woodwork came to Genoa's rescue just before the hour when Vazquez's rasping drive cannoned back off the post with Perin at full stretch.
Substitute Pinilla then spurned a gilt-edged opportunity with less than 14 minutes to play.
Bertolacci split the Palermo defence and Antonelli crossed for the unmarked Pinilla, but he somehow headed over from close range when it appeared easier to score.
