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.
Yet the turning point came midway through the second half as Menez converted from the spot after Maurizio Domizzi, who was subsequently sent off, pulled down Keisuke Honda.
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.
Milan started the game in the ascendancy and looked to press forward with every opportunity.
They almost took the lead in the 10th minute, yet Giacomo Bonaventura just failed to connect with a superb cross from Stephan El Shaarawy.
Karnezis came to the fore seven minutes later, with a stunning save to deny Rami.
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.
However, a breakthrough finally came in the 65th minute when Menez sent Karnezis the wrong way from the spot after Domizzi had hauled Honda to the ground.
The referee showed his second red card in quick succession soon after, giving Essien his marching orders for a mere shoulder charge.
Any home nerves were quickly extinguished by the excellent Menez, though.
The Frenchman cut in from the left wing with 15 minutes remaining and, having been afforded too much space by some woeful defending, smashed an unstoppable effort into the bottom-left corner.
Udinese almost pulled a goal back when Alexandre Geijo hit a powerful volley wide in an exciting finale that saw Milan spurn a couple of golden chances to add further gloss to the scoreline.
