The victory rescued an otherwise poor season for the 1991 European Cup winners, who saw bitter city foes Partizan clinch a record fifth successive league title last month with three games to spare.
Their nerves showed straight from the kick off as Marko Pavicevic's second-minute shot from 12 metres cannoned off the underside of the crossbar after an error by Red Star defender Nikola Maksimovic.
Borac, who have been relegated to the second division, squandered another gilt-edged chance when Jovan Radivojevic missed a gaping goal after losing his footing, having rounded Red Star keeper Boban Bajkovic.
Brazilian midfielder Evandro then struck against the run of play, steering home a flick-on after a miscued cross from the left which allowed Red Star's Croatian coach Robert Prosinecki to breathe a sigh of relief.
Striker Darko Lazovic twice came close against his former club in the second half before Colombian forward Cristian Borja took a loose ball in his stride and side-footed it past the keeper to send 15,000 Red Star fans into raptures.
Jubilant supporters invaded the pitch after the final whistle and stripped many of the players of their kits to celebrate Red Star's first trophy under Prosinecki, who took over the debt-ridden club in 2010.
It was Red Star's 24th national cup to add to 25 league titles and Prosinecki, a key player of the 1991 generation which tasted European glory, was upbeat after lifting the silverware.
"It was a spectacular final thanks to a jam-packed stadium, the fantastic atmosphere lifted us after a shaky start and I can only salute my players for finishing the season on a high," the 43-year old told a news conference.
"Winning the cup will be a real impetus for us next season, when this youthful team full of talents should mature and have a go at the league title.
"I am really delighted that we handled the pressure well and overjoyed with my first trophy here as coach."
