Chiesa, the 21-year-old Italy forward, sliced Roma open at will, hitting the post early on and striking twice in the first 18 minutes.
Aleksandar Kolarov pulled one back for the visitors but Luis Muriel restored Fiorentina's two-goal lead before the break.
Eusebio Di Francesco's Roma completely collapsed in the second half with Chiesa completing his hat-trick, Giovanni Simeone scoring twice and Marco Benassi also finding the net.
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Substitute Edin Dzeko was sent off for the visitors after arguing with the referee to complete a woeful night for the visitors.
Stefano Pioli's Viola will face holders Juventus or Atalanta, who play later on Wednesday, in a two-legged semi-final.
(Reporting by Rik Sharma; Editing by Toby Davis)
