The Serbia midfielder, who has made a habit of scoring long range efforts, let fly from 25 metres just before half-time with a shot that swerved into the top corner of the net.
Stankovic also had a second-half appeal for a penalty turned down when the ball hit Daniele De Rossi's arm in the box.
The win was a boost for Inter and coach Leonardo after they were dumped out of the Champions League as holders in the last eight by Schalke 04 and lost 2-0 at Parma on Saturday to all but surrender their Serie A title after five straight scudettos.
Roma, who visit Inter for the second leg on May 11, should have opened the scoring after 10 minutes but forward Mirko Vucinic missed a sitter as they went on to lose their second successive game since a U.S. consortium agreed a takeover.
Inter were without injured forward Samuel Eto'o but included right back Maicon, who was mysteriously left out of the Parma game with Leonardo refusing to give a reason. Thiago Motta, also dropped for the Parma defeat, was not in the squad.
Inter and Roma have clashed in five of the last six Italian Cup finals with Inter winning last term on the way to a treble.
Serie A leaders AC Milan, bidding for their first league and cup double, host Palermo in the second semi-final first leg on Wednesday.
