In a clash of two Italian giants, first-half goals from Emanuele Giaccherini and Alessandro Del Piero were enough to book Juve a place in the last four.
Giaccherini gave Antonio Conte's side a sixth-minute lead when he latched onto Antonio Barzagli's through pass before slotting past Maarten Stekelenburg in the Roma net.
Del Piero then doubled the hosts' advantage when he struck a trademark long-range strike from 25 yards following a headed layoff from Marco Borriello, who is on loan at Juve from Roma.
The visitors' chances were effectively ended when Eric Lamela was sent off for an off-the ball incident with Giorgio Chiellini.
A miserable night for Roma was made worse when defender Simon Kjaer diverted Fabio Quagliarella's cross into his own net in second-half stoppage time.