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The victory kept former European Cup winners Red Star on course for a domestic double with a decisive league derby against champions and city rivals Partizan to play on Saturday.
Red Star fans, close to their home turf, poured into Partizan's stadium for the Cup final and they were in raptures after Jevtic's 16th-minute opener.
The striker rose above his markers to head home Vladimir Bogdanovic's corner at the near post, after Vojvodina goalkeeper Zeljko Brkic had kept out Nikola Lazetic's effort.
Serbia striker Dragan Mrdja had several near misses for Vojvodina before Brazilian midfielder Cadu doubled Red Star's lead on the hour from a superb breakaway move engineered by Jevtic and strike partner Dejan Lekic.
Lekic headed down a lofted clearance from Red Star's penalty box and Jevtic delivered a defence-splitting pass into Cadu's path, leaving the diminutive forward with a simple finish.
It was one-way traffic after Cadu's strike, as Red Star were denied by the woodwork and desperate defending by Vojvodina before substitute Milos Trifunovic headed home the third from Bogdanovic's cross.
