Belgian international Mertens did the damage on the break, beating three players to drive inside the penalty area and smash his first Napoli goal into the bottom corner.
When play resumed after the interval Fiorentina once again dictated play, but had two penalty appeals turned down as Cuadrado and Rossi were both booked for simulation.
Rossi then forced Reina into a smart stop as his shot was diverted on its way towards the top corner, with Napoli sending on Marek Hamsik and Lorenzo Insigne for fresh legs in attack.
Ten minutes from time Napoli had a man sent off as Christian Maggio picked up his second booking, but Rafael Benitez's side held firm for a victory which sees them stay second in Serie A.
In stoppage time the referee issued another red card in controversial circumstances after Cuadrado went down under a trip in the box from Gokhan Inler, only to receive a second yellow card as La Viola lost their unbeaten home record.