Signed on a free transfer from Manchester United in the off-season, Vidic headed his new club into the lead at the end of the first half in Philadelphia on Saturday.
Yuto Nagatomo added a second for Walter Mazzarri's men midway through the second half as Inter moved, at least temporarily, top of Group A in the pre-season competition.
Their progression to the final is dependent on United's result against Real Madrid, with Louis van Gaal's men needing to avoid defeat in 90 minutes to advance to the final against the winners of Group B.
Roma had their moments in the match, not least of all when Mattia Destro was guilty of a glaring miss shortly after the break when completely unmarked at close range with only goalkeeper Samir Handanovic to beat.
As a result, Roma's participation in the tournament ended, Rudi Garcia's side having taken three points from their three matches, courtesy of a 1-0 win over Real.
Juan Iturbe had the first effort on target with a weak shot from 25 yards that was comfortably held by Handanovic in the ninth minute.
Roma suffered a blow three minutes later when defender Leandro Castan hobbled off with what looked like a thigh injury, to be replaced by Mehdi Benatia.
Zdravko Kuzmanovic then stung the palms of Lukasz Skorupski with a thumping volley from just inside the area.
Skorupski was called into action once more in the 34th minute as Mauro Icardi got on the end of Jonathan's inviting cross, but his header lacked sufficient power to trouble the goalkeeper.
Radja Nainggolan threatened for Roma four minutes later, Handanovic making a meal of his low shot from 30 yards before gathering at the second attempt.
In first-half stoppage time Vidic met Dodo's left-wing free-kick and glanced in a header that Skorupski could do little about.
Roma ought to have been level two minutes after the restart when Destro somehow headed wide of the target from five yards following a superb cross from Alessandro Florenzi.
Nainggolan's shot from distance fizzed wide of the left-hand post in the 58th minute, before the likes of Francesco Totti, Ashley Cole and Seydou Keita were all introduced off the bench by Garcia in a bid to get back into the match.
In the 69th minute, Nagatomo made the game safe for Inter. The Japan international found himself all alone at the back post and thumped a first-time finish in off the underside of the crossbar with Skorupski left helpless.
Roma did carve out one chance with five minutes remaining but Juan Carrizo, brought on for Handanovic, stood tall to deny Antonio Sanabria as the Paraguayan bore down on goal.
