Udinese were the first to threaten as Allan Loureiro found space just outside of the Juventus box, but he squandered the chance having blasted high and wide.
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Juventus almost took the lead in the eighth minute when Claudio Marchisio brought the ball down with a neat touch before laying it off to Carlos Tevez, but the Argentine striker's shot was saved by Zeljko Brkic.
Tevez was in the thick of the action again soon after as he tried to prod home from close range, although Danilo did superbly to get back and scramble the ball away for a corner.
Udinese created the first chance after the break as they almost scrambled the ball over the line from a corner, before Andrea Lazzari unleashed a stinging effort that was kept out by Buffon.
Juventus looked to take control of the game in the latter stages of the second half and Tevez almost put them ahead, but he was again denied by the outstretched Brkic, who kept the score level with 20 minutes to go.
Brkic was almost beaten in the 85th minute as Tevez bore down on goal; however, the keeper did just enough to encourage the striker to chip the ball, which sailed over the crossbar.
Fabio Quagliarella got in on the action in the 87th minute as he unleashed a stunning shot, which looked headed for the bottom corner, but again Brkic got across to tip the ball wide.
However, the goalkeeper's efforts were to prove pointless as Juventus grabbed a winner in the dying embers of the game when Llorente knocked the ball over the line 20 seconds into stoppage time to earn his side three points.