Saturday's game was put back by 45 minutes as Napoli fans protested in reaction to disturbances in the Italian capital before the match.
Flares and missiles were left off in Stadio Olimpico and captain Marek Hamsik spoke to furious Napoli supporters before the game finally got under way.
When the players eventually took to the field following a frustrating hold-up, it was worth the wait as Napoli made a blistering start.
Insigne had been criticised by Napoli fans earlier in the season, but having only previously scored five goals all season, he silenced his doubters by scoring twice in the first 17 minutes.
Vincenzo Montella's side then finally got going and Juan Vargas' first goal since February halved the deficit in the first half, but despite Giuseppe Rossi's return from injury as a substitute and Gohkan Inler's late red card, Fiorentina were unable to break Napoli down.
Substitute Dries Mertens sealed the victory with a third goal in stoppage time as Napoli secured a second Coppa Italia success in three years and silverware for coach Rafael Benitez in his first season in charge, while Fiorentina's wait for a first trophy since 2001 goes on.
Napoli's leading scorer Gonzalo Higuain, passed fit after recovering from a shin injury, somehow failed to find the back of the net after a corner caused panic in the Fiorentina defence early on.
Benitez's team were in front just 11 minutes in, though, when Hamsik picked out Insigne with an incisive pass and the forward found the far corner of the net with a measured right-footed finish.
Fiorentina were in disarray and Insigne took advantage with a second goal six minutes later.
Higuain was the architect, racing away down the right and cutting the ball across for the onrushing Insigne, whose left-footed strike deflected into the net.
It was hard to see a way back for Montella's team, such was Napoli's dominance, but they were given a lifeline when the leaders were caught napping and Vargas fired home with his left foot 28 minutes in.
Alberto Aquilani then had the ball in the back of the net on the stroke of half-time, but the Fiorentina midfielder's effort was ruled out for offside.
The Florence club were resurgent and they started the second half on the front foot.
Napoli skipper Hamsik was replaced by Mertens just after the hour mark after picking up a knock to his right shin and Goran Pandev replaced Higuain 20 minutes from time as Benitez looked to close out the win.
Fiorentina probed and Pepe Reina tipped over Matias Fernandez's long-range drive, and there was then a huge roar from Fiorentina's fans when Rossi came on with 19 minutes remaining following four months out with a knee injury.
Pandev ought to have settled it when he raced clear but Neto denied him, and Inler's second yellow card for a foul on Josip Ilicic made for an even tenser finale.
Ilicic missed a golden opportunity to equalise six minutes from time before Mertens made sure of the victory with a clinical left-foot finish two minutes into time added on.