In the day's other match Cagliari came from a goal down to beat 10-man Catania 2-1 on their return to the Sant'Elia stadium after an 18-month absence.
Birsa gave injury-ravaged Milan, who were without a host of players including striker Mario Balotelli, their third league win of the season following a 22nd-minute counter attack.
Despite a far from convincing display during which Brazilian Kaka made an uneventful 20-minute cameo appearance, Massimiliano Allegri's side leapfrogged Udinese into eighth place with 11 points from eight games.
Cagliari joined Udinese on 10 points after a superb Victor Ibarbo goal and substitute Mauricio Pinilla's close-range finish six minutes from time secured a dramatic victory in front of 4,800 delirious fans in Sardinia.
Gonzalo Bergessio had given Catania the lead with a powerful header from a fifth-minute corner and they then had Nicola Legrottaglie sent off five minutes before halftime.
Ibarbo also missed a penalty in the 41st minute but Cagliari's win was the perfect postscript to a stadium saga which meant the club were barred from playing at two grounds before being forced to play their home games 1,000-km away in Trieste.