The hosts held out until just past the hour mark when the Spaniard volleyed his side in front, before the ex-Athletic Bilbao man teed up his strike partner to settle matters
The home side battled bravely for an hour until the Spanish striker volleyed home from Paul Pogba's cross, before teeing up Tevez to drill home a second with fifteen minutes left to play.
The result sends defending champions Juventus to the summit for the time being, with unbeaten Roma not playing until Monday.
Antonio Conte's men have now kept five consecutive clean sheets in the league and are two points ahead of Rudi Garcia's charges.
Chilean midfielder Arturo Vidal, who had to gain Fifa clearance to play after pulling out of both of his country's recent friendlies, stepped into a makeshift Juventus defence, with Stephan Lichtsteiner, Andrea Barzagli and striker Sebastian Giovinco all out injured, while Leonardo Bonucci and Angelo Ogbonna were suspended.
Livorno made four changes to the side that lost 2-0 at Inter last time out, with Andrea Coda, Leandro Rinaudo, Leandro Greco and Innocent Emeghara coming in for Giuseppe Gemiti, Nahuel Valentini, Alfred Duncan and the suspended Paulinho.
After a quiet opening 20 minutes, Llorente had the visitors' first clear-cut opportunity, heading over the crossbar from a Simone Padoin cross.
Emeghara pulled the trigger from the edge of the penalty area at the other end five minutes later, but his effort was deflected high and wide for a corner.
Paul Pogba forced Livorno goalkeeper Francesco Bardi to tip over his crossbar on the half-hour mark with a powerful drive from 20 yards, but Juve were struggling to find their flow against a disciplined home defence.
And the hosts almost broke the deadlock three minutes later, but Emeghara's chipped attempt drifted wide of the target amid suspicions of offside.
Kwadwo Asamoah fancied his chances from distance, but the Ghanaian saw his effort pushed around the post by Bardi eight minutes before the interval.
Greco went into the referee's book after 53 minutes for simulation in the penalty area after a collision with Vidal, but replays suggested it was a harsh decision.
Juve went ahead in the 63rd minute as Llorente met a Pogba cross from the right to send a right-footed volley into the top right-hand corner of the net from the centre of the penalty area.
Tevez drilled just wide of the left-hand post as the visitors went in search of a second, but Livorno were holding firm as they clung onto the belief that they could still take something from the encounter.
However, Tevez killed off any lingering hopes that remained for the hosts as he fired home a low effort from 16 yards after good work in the area by Llorente, before Pogba struck the post in the dying minutes as Juve finished strongly.