The gap at the summit that was partially closed by Kalidou Koulibaly's late winner in Turin last week has now been restored after the same man saw red early on to set the tone for a miserable outing for his side.
A dramatic victory at Inter for champions Juve piled the pressure on Napoli to deliver three points from a tricky trip on Sunday, but their defence never recovered from Koulibaly's setback.
Simeone, who drew the costly foul, deservedly netted the opener after 34 minutes and then completed his hat-trick as Fiorentina's dominance continued after the break.
The Partenopei have regularly relied on late goals and fightbacks in the run-in, but neither were forthcoming and Sarri's side now require something of a miracle in the final three weeks of the campaign to end their 28-year wait for a title.
Koulibaly was dismissed after just six minutes when he mistimed a challenge on Simeone as the forward charged into the area.
Referee Paolo Silvio Mazzoleni initially awarded a yellow card and a penalty, but replays showed the foul to have taken place marginally outside the box, prompting Koulibaly's dismissal and a Cristiano Biraghi free-kick that Pepe Reina blocked.
Sarri sacrificed Jorginho for centre-back Lorenzo Tonelli, yet the visitors were beaten again by the same Simeone run in behind and this time he found the net, sliding a finish beyond Reina.
Reina clung onto Riccardo Saponara's low drive and then smothered efforts from both Marco Benassi and Federico Chiesa at close quarters, with Napoli's defence still scrambling.
1 - Napoli have attempted just one shot against Fiorentina so far; their joint-fewest in the first half of a game in Serie A this season (also previously against Fiorentina). Blocked. #FiorentinaNapoli— OptaPaolo (@OptaPaolo) April 29, 2018
A sublime Jose Callejon pass picked out Dries Mertens for a volley straight at Marco Sportiello following the restart, but Fiorentina continued to create chances and Reina parried Chiesa's bobbling shot.
Simeone's second arrived with just under half an hour remaining. Although Reina was blocked smartly as a corner struck Elseid Hysaj, the home forward on hand to stab the rebound past the helpless goalkeeper.
Napoli's response was negligible as Fiorentina instead ended the game chasing a third.
And with just seconds of stoppage time remaining it arrived, Simeone bursting clear to finish left-footed and clinch the match ball as the visitors' valiant Scudetto fight wilted in the Tuscan sun.