Mistakes gifted Inter the crucial goals in the first half, before the Udinese challenge faded to allow the visitors to cruise to the win.
Icardi punished Udinese for not scoring while on top as Bruno Fernandes lost possession to play in the striker, who displayed his composure to fire home the opener.
Handanovic came to the rescue for the visitors once again to get a crucial touch on an effort from Antonio Di Natale after the Udinese captain had been gifted space in the box.
Another mistake gifted Inter their second goal when Maurizio Domizzi hesitated over a pass back to his goalkeeper, which allowed Jovetic to sprint clear and chip the ball over Karnezis.
Inter quickly improved after doubling their lead, and the league leaders managed the game with control either side of the interval.
Adem Ljajic could have killed the game off on the hour mark when Icardi headed a cross into his path, but the Serbian could only poke his effort wide of the post.
A clinical edge to Udinese's play could have made the evening much more difficult for Inter, who pulled further ahead late on when Icardi created room for a shot to leave Karnezis powerless to prevent the ball finding the top corner.
There was still time for Inter to add gloss to the scoreline three minutes later as a counter-attack finished with Marcelo Brozovic cutting inside from the left to curl home.