Champions Juve moved on to 56 points from 21 games after Spanish centre forward Llorente met Stefan Lichtsteiner's cross to score with a smart, looping header on the hour that cancelled out Antonio Candreva's 27th minute penalty.
"These things go for and against you," said Juve coach Antonio Conte, "but I think a penalty is already enough of a punishment, and I think a penalty and a booking should have been the maximum punishment handed out.
"We want to win, whether we have 11 or 10 men. We look to cultivate a winning mentality and these boys know very well that you play to win.
"The response (to the sending off) was good. They showed themselves to be men rather than just players. I think the team showed why they are top of the league."
However, Lazio were unlucky not to win after striking the woodwork twice in the second half, first with a Klose header that keeper Marco Storari pushed on to the bar and then through Keita Balde Diao's 86th minute curling, long range strike that rebounded off the post and back into Storari's arms.
The draw means Juventus have widened the gap on second-placed AS Roma, who have 47 points and face a tricky trip to surprise package Hellas Verona on Sunday (1130 GMT).
Lazio stay ninth on 28 points and continue their unbeaten run under new coach Edy Reja, who has seen his side gain eight points from four games since returning to the dugout as the replacement for Vladimir Petkovic after the winter break.
In Saturday's earlier game Napoli left their own feint title hopes hanging by a thread after needing a late Raul Albiol goal to snatch a 1-1 draw against struggling Chievo.
The draw came after a lacklustre attacking display by Rafael Benitez's side, who were nevertheless unlucky in front of goal.
Gonzalo Higuain missed an open goal in the first half when Jose Callejon's mis-hit shot found him in front of an empty net, only for the Argentina striker to somehow put the ball out for a throw-in.
Dries Mertens hit the post twice, either side of the break, and Lorenzo Insigne struck the bar with a powerful shot soon after coming off the bench.
It was another disappointing home result for Napoli after a defeat by Parma and draws against Sassuolo and Udinese in the first half of the season slowed what many thought would be a strong bid for their first title in over 20 years.