The Frenchman scored a cool penalty and a fine solo goal to ensure Filippo Inzaghi's men triumphed in a game featuring two red cards.
Milan, who entered Sunday's game sitting seventh in the table on the back of their poor form, looked to impose themselves - only to initially run into an inspired Karnezis, who produced a string of world-class saves in the opening half.
That one of those stops, from an Adil Rami header, appeared to be made with the ball over the line only added to the growing frustration.
It quickly became 10 versus 10 when Michael Essien was given a harsh second yellow card, but Milan took the spoils as Menez struck again.
The defender rose highest to meet Bonaventura's corner at the near post, only for Karnezis to somehow divert it on to the woodwork - albeit with replays showing the ball may have been over the line.
Milan's frustration was added to significantly as El Shaarawy and Bonaventura hit wide from distance in quick succession.
Udinese came back into the game a little thereafter, although they were again indebted to Karnezis when he produced a one-handed stop to turn away El Shaarawy's cross-cum-shot.
That save paled in comparison to his next effort, however, with Karnezis getting down to his right to somehow turn away Menez's effort with his fingertips.
The second period continued in much the same vein, with the shot-stopper punching clear Essien's long-range effort.
The referee showed his second red card in quick succession soon after, giving Essien his marching orders for a mere shoulder charge.
