A win would have seen Stefano Pioli's side leapfrog both Sampdoria and Fiorentina in the table, but defeat means they remain sixth.
The key moment in the game arrived shortly before the half-hour mark.
On-loan Milan forward M'Baye Niang sprung the offside trap and raced onto a pass from Andrea Bertolacci, where he was taken out inside the area by Marchetti.
Referee Andrea Gervasoni had little option but to brandish a straight red card and replacement keeper Etrit Berisha was brought on with Danilo Cataldi sacrificed.
Berisha could do nothing about Perotti's calm spot-kick, however, as Genoa ended a seven-match winless run and climbed to just one place and two points behind their opponents in the table.
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