Matthijs de Ligt's strike moved the Old Lady to the brink of glory on the stroke of half-time, but a beleaguered Sarri was once again left seething after his side fell apart after the break.
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Fofana's brilliant individual effort in the 93rd minute did not allow Juventus to wrap up the Serie A title.
They play Sampdoria on Sunday (20:45 BST) - although could be champions before then if their rivals lose their weekend matches.
Juventus - who had failed to win in their previous three games - now only need four points in their last four games to seal the first league title of manager Maurizio Sarri's career.
"We wanted to win at all costs, we were messy and we took the game on a risky track and lost it on 93 minutes because we wanted the three points".
"In this way we took the game to a unsafe level, and after the 90th minute, we lost it".
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"In this period we are physically and mentally exhausted, and it is a problem common to all", Sarri said.
Juventus's performance against the minnows from Udine, close to the border with Slovenia, was worrying ahead of their crunch Champions League last 16 game against Lyon on August 7.
The game started with a clumsy header from Danilo, which nearly resulted in an own goal before bouncing off Wojciech Szczesny's post. A defensive header out came straight to De Ligt and he took one touch before firing powerfully into the bottom left corner.
Fofana raced from the halfway line to score in the second minute of time added on to lift Udinese seven points clear of the drop zone with three games to go, but extend the wait for a Juve team badly out of form.
"I am very happy because of these three heavy three points and also because before the game, no one would have bet on us", Troost-Ekong told Udinese TV, as cited in Complete Sports.
"It's hard to keep up mentally and physically for 90 minutes".
Sergey Milinkovic-Savic got Lazio back into the game just after the break with Ciro Immobile connecting with a Luis Alberto cross for the victor after an hour in Sardinia.
Simone Inzaghi's side ended their five-match winless run with a victory which leaves them with a slim chance of winning the title, as they sit eight points behind Juventus.