Rafael Benitez's side, who will face Fiorentina in the final, took the lead on the night and levelled the tie when Jose Callejon scored a close-range header from right back Christian Maggio's perfectly flighted cross after 33 minutes.
An attacking blitz from Napoli in the opening minutes of the second half saw Gonzalo Higuain double their lead from a corner and Jorginho net his first Napoli goal to seal their place in May's final.
Kevin Strootman compounded Roma manager Rudi Garcia's misery by being sent off in the 79th minute.
The Dutch midfielder was booked after a cynical foul on Callejon, before he received his marching orders immediately afterwards for sarcastically applauding referee Gianluca Rocchi.
Napoli will take on Fiorentina in the Italian capital after Vincenzo Montella's side came back from a 2-1 first leg defeat to Udinese to take the tie 3-2 on Tuesday.
"I'm very happy, it was a great night and a fantastic match, to win 3-0 against such a good team. We deserve to be in the final," said Higuain.
"I'm here to help the team, and I'm really happy for the match we played tonight," added the Argentine, who scored his 18th goal of the season in front of Napoli great Diego Maradona, who arrived at the San Paolo surrounded by security guards.
It was Roma's first defeat in eight games in all competitions.
"We made mistakes in the first leg and tonight. At 1-0 anything could have happened, as the tie was still in the balance," said Roma boss Garcia.
"If there was a team that should have been ahead at the end of the first 45 minutes it was us, but we weren't and that was our own fault."
For Napoli, the victory keeps them in the hunt for silverware having fallen away in the Serie A title race.
The away side started positively and could have taken the lead with Mattia Destro and Gervinho passing up chances before Callejon found the net.
They continued to threaten throughout the opening period, but the second half saw a collapse not dissimilar to their 3-0 defeat at Juventus in January, with Higuain tucking home an easy header after 48 minutes.
Brazilian Jorginho then took advantage of confusion in the dazed Roma backline.
He met a sumptuous Dries Mertens through ball to expertly flick past keeper Morgan De Sanctis to complete the victory and give Napoli a chance of winning their second Italian Cup in three seasons.
Image: Gonzalo Higuain and Josè Manuel Reina of Napoli celebrate
Photograph: Giuseppe Bellini/Getty Images
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