Inzaghi's side headed into the clash at San Siro on the back of a five-game winless run, but took the spoils thanks to Jeremy Menez's second-half double in a game featuring two sendings off.
Udinese's Maurizio Domizzi was the first to go, shown a straight red card having hauled Keisuke Honda to the ground in the area.
Menez converted the subsequent penalty and, after Milan's Michael Essien had been given a harsh second booking, added a second with a fine solo goal.
The victory delighted Inzaghi, who told Sky Sport Italia: "We should've been 3-0 up at the break and it was still goalless, so I think overall it was Milan's best performance of the season.
"My objective this term is to get the team playing good football, for the fans to be smiling in the stands and to make our president happy."
Inzaghi refused to be drawn on a controversial refereeing performance from Paolo Valeri, which began when Adil Rami was denied a goal when the ball had seemingly crossed the line.
He added: "Let's just stick to how we played, which is the real importance of football. We dominated from start to finish, so a couple of incidents won't change that.
"I only have two ways of getting slowly back to where we were and those are good football and hard work."