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Ante Rebic's superb 13th-minute opener appeared to put Milan on course for a routine win, but they made heavy weather of things thereafter.
Former Milan loanee Mattia Destro levelled before half-time and the Rossoneri required an own goal from Gianluca Scamacca to claim maximum points.
Inter can stretch their lead at the top back to 11 points by beating Napoli later on Sunday, but Milan's slim hopes are still alive, while this result is another step towards making sure of Champions League qualification.
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Genoa goalkeeper Mattia Perin had not been tested in the opening 12 minutes but he was left with no chance when Rebic unleashed a stunning volley on the turn to find the bottom-right corner in style after the visitors failed to clear having initially blocked Theo Hernandez's effort from a free-kick.