Alessandro Matri scored both of Juve's goals with Antonio Floro Flores equalising for Udinese early in the second half.
Matri missed two further chances, but Udinese did not just make up the numbers as they searched for goals themselves, justifying their third place in the table.
Despite the heavy snow falling in Piedmont, the pitch in the Juventus Stadium was in pristine condition with a slick surface making for a fast pace with both sides keen to attack.
It was a case of third time lucky for Matri when he finally put Juventus in front in the 42nd minute.
He missed two one-on-one chances in the second and 23rd minute with Udinese goalkeeper Samir Handanovic coming out on top both times to deny him.
Not wanting to be outdone, at the other end, Gianluigi Buffon made a fine one-handed save to deny Pablo Armero in the 16th minute.
But it was Juve who were the most likely to score and, after Stephan Lichtensteiner shirked the chance to shoot from a good position in the 34th minute, Matri struck.
Marcelo Estigarribia's cross from the left was headed towards goal by Fabio Quagliarella.
Handanovic got a hand to it, but only managed to lay it on a plate for Matri, who made no mistake this time from close range.
Udinese started to take more risks early in the second half with Antonio Di Natale shooting just wide from 20 yards out in the 53rd minute.
The visitors were particularly dangerous on the counter attack and that is how they levelled in the 56th minute with Mauricio Isla leading a three-against-two attack before feeding Floro Flores, who picked his spot in the far corner of Buffon's goal.
A four-against-three break minutes later culminated in Di Natale lifting his close-range shot over the crossbar.
Juve were under pressure, but they found the best way out of it by regaining the lead in the 62nd minute.
Claudio Marchisio touched the ball through to Matri, who held off a defender and turned to shoot into the opposite corner of Handanovic's goal.
It was enough to earn Juve all three points and take them four points clear of AC Milan at the top of Serie A.