Pirlo was injured in the sixth minute of Juve's last-gasp 1-0 win after being on the receiving end of a crunching Andrea Lazzari tackle.
He continued on playing but was withdrawn eight minutes later, with Serie A champions Juve confirming on Monday that he had suffered ligament damage to his right knee.
"Andrea Pirlo, who picked up a knock not long after kick-off in yesterday's fixture between Juventus and Udinese, has undergone medical tests this afternoon," a club statement read.
"The examinations at Turin's Fornaca Clinic have revealed second-grade damage to the medial collateral ligament of his right knee."
Injuries of this nature can often sideline players for up to two months, but Juve will be hopeful midfield maestro Pirlo can return after the mid-season break ends on January 6.
Juventus - who have won the last two Serie A titles - top the table by three points after Fernando Llorente's injury-time winner against Udinese.
