Two goals from the former Nerazzurri striker inspired the Giallorossi to come from behind to set up a mouth-watering showpiece at the expense of Andrea Stramaccioni's men
The Nerazzurri held an advantage in the tie for the first time when Jonathan, who was a late addition to the squad following an injury to Esteban Cambiasso, broke the deadlock in the first half. However, Mattia Destro returned to haunt the club who rejected him as a youngster with two second-half goals to turn the tide.
Vasilis Torosidis grabbed a spectacular third for the visitors to set up a date in the final with Rome-rivals Lazio before Ricardo Alvarez netted a late consolation strike.
Disappointment at the weekend's Serie A defeat to Cagliari was evident in the Inter ranks from kick-off and they almost took another blow after just three minutes. After initiating a lovely, flowing move down Roma's left, Marquinho was only denied a simple finish from Alessandro Florezni's cross by the outstretched arm of Samir Handanovic.
The Nerazzurri soon awoke and in the 22nd minute they took the lead with a goal of extreme quality. Following a give-and-go with Alvarez, Jonathan stormed towards the area and played the ball into Tommaso Rocchi. The veteran striker rolled it straight back into the path of the Brazilian who slid his shot into the bottom corner.
Moments later the advantage could have been doubled but Rocchi's connection with Ezequiel Schelotto's cross was poor, leaving Maarten Stekelenburg with a comfortable save.
Inter were sitting deep in their own half when not in possession and this allowed the Giallorossi plenty of space. As a result, a searing shot from Marquinhos whistled just wide after it took a deflection off Walter Samuel.
The visitors maintained a high level of pressure but found their attempts to draw level thwarted by the in-form Handanovic. The Slovenian keeper denied Destro from point-blank range in the 34th minute and then turned a shot from Florenzi around the post on the stroke of half-time.
Roma did not let up after the break and after Francesco Totti and Torosidis had both gone close, they finally equalised in the 56th minute. Erik Lamela's perfectly-weighted through-ball sent Destro in behind the defence and the former Inter player dinked the ball over Handanovic and into the net.
Inter's shaky defence has frequently been their undoing this season and it proved to be the case once more in the 69th minute. Juan Jesus failed to track the run of Federico Balzaretti, who laid Daniele De Rossi's pass across the face of goal and Destro was on hand to tap in his second.
Andrea Stramaccioni's side quickly fell apart and five minutes later the tie was well and truly put out of their reach. A magnificent, looping strike from the corner of the box by Torosidis compounded the hosts' misery.
Alvarez managed to pull one back for the hosts with a low shot inside the near post that was no less than he deserved, though it proved to be no more than a consolation.
Thus it is Aurelio Andreazzoli's men who secure a space in the final where they will face arch-rivals Lazio, who eliminated Juventus in the other semi-final.