The Sunday night meeting, which somehow ended 1-0, might have been a goalfest if not for remarkable displays of goalkeeping from Gianluigi Buffon and his visiting counterpart Zeljko Brkic.
The veteran Juve stopper was particularly impressive in steering the reigning champions through a sixth league game without conceding a goal.
Brkic stood firm in the final half hour as Antonio Conte's men steadily turned the screw but Spain international Llorente burst his bubble with a header in the first minute of three added on at the Juventus Stadium.
The three points gave Juve a comfortable cushion on previous runaway leaders Roma, who were earlier held in Atalanta. The title rivals meet in the capital city on January 6.
It was a sixth straight league win for the resurgent Bianconeri while an eighth defeat of the season sees Udinese slide into 13th place.
Juventus were, of course, cheered by news of Roma's failure to beat Atalanta - a meeting where the Giallorossi struggled to force a 1-1 draw - and duly pushed hard for an early goal.
Some great ball control and deftness of touch soon saw Claudio Marchisio send Carlos Tevez through on goal but Brkic repelled the Argentinian's first-time effort.
Roberto Pereyra brought a regulation save from Buffon at the other end before Juve suffered an unexpected setback.
Veteran playmaker Andrea Pirlo had taken a knock in the opening exchanges and could not continue beyond the 14th minute, when he was replaced by Paul Pogba.
Llorente tried to lift the hosts' spirits with a lung-busting run that brought him into the box. The Spaniard teed up Tevez but the finish was deflected away for a corner.
The 29th minute saw Udinese gifted a golden opportunity to take a shock lead.
Leonardo Bonucci, normally unflappable in defence, was guilty of committing a terrible error in front of Buffon, who somehow clawed Antonio Di Natale's clever lob away with his fingernails.
The two Serie A stalwarts would go head-to-head when play resumed after the break, too.
This time, after Bruno Fernandes served Di Natale with a pinpoint cross, the striker was left disbelieving as Buffon blocked his finish from point-blank range.
Juve's long-serving stopper continued to steal the limelight, even if Di Natale did his best to upstage him with a skilful overhead flick and run before feeding Andrea Lazzari down the left channel.
The Fiorentina loanee struck an angled drive from the edge of the box but Buffon defied his 35 years by athletically palming the ball over the bar at full stretch.
Brkic kept his own sheet clean when Tevez and Llorente caused a goalmouth scramble in the Udinese box, but by now Juve were looking ominous and growing in confidence.
Coach Conte introduced Fabio Quagliarella to make a late impact. The striker was, somewhat unusually, shown a yellow card for handball within only 20 seconds of his arrival on the pitch.
The acrobatic hitman presented a serious threat to Brkic, though, and proved as much with a fine right-footed drive three minutes from time, the Serbian stopper impressive in making a crucial save.
Neither Brkic nor his Udinese team-mates would not escape Turin unscathed as Llorente shattered their resolve with a header winner seconds after the 90-minute mark had ticked by.
Stephan Lichtsteiner provided the cross for the rangy Spaniard, who rose highest to glance the ball down and into the goalmouth.