Genoa's Serbian forward Nikola Ninkovic had opened the scoring after 11 minutes before Paletta was dismissed in the 56th. Kucka put the ball into his own net 10 minutes from time with Leonardo Pavoletti wrapping up the victory late on.
Vicenzo Montella's Milan stay third on 19 points from 10 games, two behind champions and leaders Juventus, who they beat 1-0 on Saturday, while Genoa jump five places to seventh on 15.
Juve host Sampdoria on Wednesday when second-placed AS Roma, who are level on 19 points with Milan, visit Sassuolo.
The 21-year old Ninkovic, often in trouble at his former club Partizan Belgrade for failing to control his temper, finally showed a glimpse of his vast potential when he fired Genoa ahead with a diving header for his first Serie A goal.
Ninkovic, who joined Genoa in February but was immediately loaned out to Chievo Verona, was replaced by compatriot Darko Lazovic in the 54th minute moments before Milan defender Paletta was sent off for a lunging challenge on Luca Rigoni.
That gave former Red Star captain Lazovic plenty of room to operate on the right flank and from one such raid Genoa doubled their lead when Kucka diverted the Serb's teasing low cross past his own goalkeeper Gianluigi Donnarumma.
Pavoletti, also a second half substitute, made it 3-0 when he curled an unstoppable shot into the far corner in the closing stages as Milan fell apart at the back.
The seven-times European champions looked a pale shadow of the side that impressed against Juve, showing they are still very much a work in progress under new coach Montella.
(Writing by Zoran Milosavljevic; Editing by Ken Ferris)
