Holding a 2-1 lead from the first match, Roma went behind to a superb Jonathan goal before former Inter trainee Mattia Destro scored twice in 14 second-half minutes to put the tie firmly in Roma's hands.
Vassilis Torosidis hit a wonderful third for the visitors and Ricardo Alvarez rounded out the scoring with an 80th-minute consolation for an Inter side missing several first-choice names.
Roma started brightly, working Samir Handanovic twice in the first five minutes - once when he pawed away Alessandro Florenzi cross-shot and moments later when he read Francesco Totti's free-kick.
But it was Inter who went ahead after 22 minutes, Jonathan carrying the ball from midfield and exchanging silky one touch passes with Tommaso Rocchi and Alvarez before sweeping the ball into the bottom corner.
Marquinho went close to an equaliser 10 minutes later but his deflected effort squirmed wide.
Handanovic made two fine saves to keep Roma out before half-time, bravely blocking a Destro shot at his near post and then denying Florenzi.
Roma had their leveler in the 55th minute, Marquinho springing the offside trap, Destro timing his run to perfection and then delicately lifting over Handanovic to leave Walter Samuel attempting a goalline clearance in vain.
Inter almost found the ideal response but Rocchi just failed to pick his run at the end of a neat counter-attack in the 68th minute.
Instead, Roma broke and clinically made it 2-1, as Daniele de Rossi and Federico Balzaretti combined to tee up Destro, who provided a simple finishing touch from close range.
What followed five minutes later was a stunning solo effort from Torosidies, from a seemingly innocuous, position the full-back hooked the ball over his shoulder, spun his marker and took a touch on his chest before lashing into the top corner.
It was a goal worthy of having the final say but Inter had one final moment to cheer as Alvarez cut in from the right and wrong-footed Maarten Stekelenburg at his near post.