Spaniard Llorente struck the opening goal on his first start for Carlo Ancelotti’s side before Lorenzo Insigne doubled their lead with a penalty before the break.
A Fernando Llorente double helped Napoli move into third place in Serie A with a 4-1 win at Lecce on Sunday, while Edin Dzeko scored a last-gasp winner as Roma beat Bologna 2-1.
Spaniard Llorente struck the opening goal on his first start for Carlo Ancelotti's side before Lorenzo Insigne doubled their lead with a penalty before the break.
Fabian Ruiz's curling strike made it three before Marco Mancosu pulled one back for Lecce from the penalty spot, but Llorente made sure of the points with a tap-in.
In Bologna, Roma defender Aleksandar Kolarov went from hero to zero in the space of five second-half minutes as he scored a superb free-kick before conceding a penalty to gift Nicola Sansone the equaliser.
Roma's Gianluca Mancini was shown a second yellow card with five minutes remaining but Dzeko powered home a header from point-blank range in the dying seconds to send the capital club fourth.
Napoli are third on nine points, one behind Juventus and three off leaders Inter Milan, while Roma are one point behind them and Bologna occupy fifth place on seven points.
Elsewhere, Franck Ribery opened his account for Fiorentina but his new club were denied their first win of the season as Atalanta fought back from two goals down to draw 2-2.
Ribery scored a sensational second-half volley after Federico Chiesa's opener, but Atalanta pulled one back through substitute Josip Ilicic with six minutes remaining before Timothy Castagne volleyed in a 95th-minute equaliser.
Fiorentina are now without a win in their last 18 Serie A games, the worst run the club has been on since going 21 matches without victory in 1937/38.
Sampdoria earned their first points of the season with a 1-0 win over Torino, while Francesco Caputo scored twice in Sassuolo's 3-0 victory against SPAL.
Former Tottenham Hotspur striker Llorente earned a starting spot for the first time since joining Napoli on a free transfer on deadline day.
The 34-year-old scored in the 2-0 Champions League victory over Liverpool during the week and was again decisive as he latched on to a deflected shot to roll in Napoli's opener after 28 minutes.
Insigne then scored from the spot at the second time of asking after goalkeeper Gabriel was pinged for coming off his line early when he saved the first effort, resulting in a retake.
Ruiz struck a sublime third after the break before Mancosu gave the home side hope from the spot, but Llorente ended any suggestion of a comeback when he pounced on Gabriel's parry to tap in his side's fourth with eight minutes remaining.
Bologna and Roma were both unbeaten going into Sunday's meeting and Dzeko's last-minute header ended an eight-match winning home run for Sinisa Mihajlovic's side.
Kolarov curled a wonderful free kick into the top corner from the edge of the area after 49 minutes to put the visitors in front. However, he then tripped up Roberto Soriano inside the box just four minutes later, allowing Sansone to send Roma goalkeeper Pau Lopez the wrong way.
Bologna threatened a shock win as Lopez was forced into a superb save on Soriano before Mancini was given his marching orders for throwing an arm in the face of Federico Santander.
Dzeko rose to head in the winner in the final minute of added time to wrap up the points.