The Serie A leaders - who had lost in the same stadium to Roma in the Coppa Italia on Tuesday - appeared to be heading for their first league defeat since October when Antonio Candreva notched from the penalty spot following the dismissal of Gianluigi Buffon, who brought down Miroslav Klose in the area.
But Llorente levelled proceedings in the second half to leave Lazio waiting for a first league triumph over Juventus since December 2003.
The second half continued in a similar fashion, and the champions deservedly equalised when a well-0worked move was finished by Llorente's fine header.
The draw ends Juve's 12-match winning league run, while Lazio have now won two and drawn two league matches since Edy Reja returned as coach earlier in January.
At the other end, Lazio almost took the lead in bizarre circumstances when Klose caught Buffon in possession and the Germany international's challenge forced the ball just wide of goal.
Buffon's night went from bad to worse when he was shown a straight red card for denying a clear goalscoring opportunity after bringing down Klose in the area after 24 minutes.
Despite that setback, Juve continued to see plenty of the ball and they created a great opportunity nine minutes before half-time when Stephan Lichtsteiner's low centre was collected by Llorente, who turned sharply in the box and forced a smart save from Etrit Berisha.
Antonio Conte's charges showed their ruthlessness on the hour mark, however, as Llorente met Lichsteiner's pin-point cross with a looping header to the far corner.
However, with 15 minutes remaining, Storari produced a fine point-blank save to deny Klose's header from Cristian Ledesma's whipped free-kick.
A near carbon copy of that set-piece then led to Storari holding well from Klose's header, and the hosts kept up their late pressure when substitute Balde Keita curled an effort against the post in the 85th minute, but both sides were forced to settle for a point.
