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Kevin Constant came close to opening the scoring with a volley in the fourth minute, but the visitors responded well, with Andrea Costa going within inches of heading past Christian Abbiati in the 27th minute.
The teams went in level at the break, but the hosts caught their opponents cold seconds after the restart, with Birsa drilling a fierce low drive beyond Junior Da Costa from just outside the area.
Robinho should have added to the Rossoneri lead three minutes later, but somehow he missed the target from close range.
The Brazilian's miss proved to be of little consequence as the home side went on to see out their second victory of the season, leaving themselves six points behind league leaders Napoli.