A Gervinho brace in the first leg gave Roma a 3-2 advantage heading into Wednesday's fixture at Stadio San Paolo but Rafael Benitez's side were superb on home soil and secured a 5-3 aggregate triumph.
Jose Callejon headed in the opener in the 33rd minute as Napoli - who have won the tournament on four occasions, most recently in 2012 – took the lead on away goals.
And strikes in the first six minutes of the second half from Gonzalo Higuain and Jorginho sealed the result as Napoli won in front of club legend Diego Maradona.
Dries Mertens scored as a substitute in the first leg and was promoted to Napoli's starting line-up for the return fixture while nine-time Coppa winners Roma made three changes from their win in the opener as captain Francesco Totti dropped to the bench.
Mattia Destro almost gave Roma the lead in the third minute from Adem Ljajic's cut-back, but Napoli goalkeeper Pepe Reina was equal to the task and made a great save from point-blank range.
In a quiet first half, Gervinho had an effort blocked while Ljajic shot over, but just as Roma began to take control, Napoli claimed the lead.
Callejon got in between Vasilis Torosidis and Mehdi Benatia and converted his free header with aplomb, nodding Christian Maggio's excellent cross to the right of Roma's ex-Napoli goalkeeper Morgan De Sanctis.
Ljajic shot at Reina from a tight angle after he combined well with Miralem Pjanic as Roma searched for a response but they were blown away just after the break.
Callejon scuffed an effort at De Sanctis after linking excellently with Higuain before Napoli scored again from the resultant corner, which was flicked on at the near post.
Higuain managed to get away from Torosidis to head in at the far post in the 48th minute and it was almost immediately followed by Jorginho's effort.
Mertens provided the third, cutting in from the left before producing a superbly timed ball that allowed Jorginho to beat the offside trap, before lifting the ball past Reina and in.
Kevin Strootman's header was ruled out for offside and Destro nodded wide after Maicon's cross as Roma battled for a lifeline.
Strootman's night went from bad to worse in the 79th minute as he was sent off for fouling Callejon, which he followed up by kicking the ball at the player and sarcastically clapping the referee.
His actions summed up Roma's night, as Napoli were simply too good, setting up a meeting with Fiorentina in the showpiece.
