Borini, the man whose goal secured derby victory over Newcastle earlier this season, nervelessly converted to condemn the visitors to three defeats in a row for the first time since May 2001, and the result will send Sunderland to Old Trafford on January 22 with high hopes of reaching their first major final since 1992.
The Black Cats, who remain rooted to the foot of the Barclays Premier League table, had taken the lead at the end of the first half as veteran Ryan Giggs suffered a night to forget on Wearside.
The 40-year-old proved a central figure, but not in the way he might have wished as his luck deserted him.
Adnan Januzaj's 38th-minute strike, which would have given United the lead, was chalked off because the Welshman was offside when the 18-year-old's initial shot hit him in front of goal, and it was Giggs who handed the home side the advantage in injury time.
Giggs found himself patrolling at the far post when Wes Brown drilled Sebastian Larsson's free-kick in his direction and as Phil Bardsley attempted to dispatch it, he turned it into his own net.
The visitors dragged themselves back into the game when returning central defender Nemanja Vidic struck with a towering 52nd-minute header, but the drama was far from over.
Substitute Adam Johnson lured Tom Cleverley into a ill-judged challenge on the edge of the box and, after consulting his assistant, referee Andre Marriner pointed to the spot to hand Borini his big chance.
The tie, of course, is far from over and United, who were watched by their former manager Sir Alex Ferguson and director Sir Bobby Charlton, may approach the second leg confident of clawing back the deficit.
However, they have already lost five times at Old Trafford this season - to West Brom, Everton, Tottenham, Newcastle and Swansea - and Sunderland will travel dreaming of a first trip to the League Cup final since 1985.
Despite their lowly league standing, the Black Cats started confidently against a United side whose form had been indifferent leading into the game.
Debutant Marcos Alonso provided a useful outlet down the left, while Ki Sung-yueng's invention from central midfield kept the United rearguard on its toes.
But in truth, the home side never really threatened until the closing stages of the half, and it was the Red Devils, who took time to get going, who had the greater threat.
After Borini and Larsson had gone for goal from long distance at the other end, the visitors started to make an impact and nearly took the lead when Michael Carrick picked out Giggs in space and the Welshman was allowed to make ground before unleashing a shot which clipped Bardsley and crashed against the crossbar.
With Carrick starting to pull the strings and full-back Patrice Evra and midfielder Januzaj finding space, United started to find their rhythm and Bardsley had to head away Januzaj's curling 33rd-minute shot as it arced toward the top corner.
The United teenager thought he had made the breakthrough five minutes later when, after his initial effort had hit Giggs in front of goal, he rammed home the rebound.
However, an offside flag correctly ended his celebrations with Giggs having strayed beyond the last defender when he made his unwitting contribution.
Manager David Moyes' disappointment was to increase significantly two minutes into injury time at the end of the first half when former United defender Brown turned Larsson's free-kick across goal and Giggs bundled it under his own net under extreme pressure from another former club-mate, Bardsley.
It took the visitors just seven minutes of the second half to restore parity, and they did so in emphatic style when Cleverley floated an enticing corner to the far post and Vidic climbed high above Brown and John O'Shea to power a header home.
Sunderland, for whom Larsson had forced a fine save from David De Gea minutes earlier, refused to succumb and they regained their lead when Cleverley tripped Johnson, with referee Marriner awarding a penalty on the advice of assistant Peter Bankes. United were furious but replays suggested it was rash from Cleverley and suitably punished.
Borini sent the goalkeeper the wrong way from the spot to put United on the back foot once again, and although the visitors threw men forward as time ran down, they rarely threatened to square the tie.
If you have a complaint about the editorial content of the Belfast Telegraph or Sunday Life then contact the Editor here. If you are not satisfied with the response provided then you can contact the Independent Press Standards Organisation here
COMMENT RULES: Comments that are judged to be defamatory, abusive or in bad taste are not acceptable and contributors who consistently fall below certain criteria will be permanently blacklisted. The moderator will not enter into debate with individual contributors and the moderator’s decision is final. It is Belfast Telegraph policy to close comments on court cases, tribunals and active legal investigations. We may also close comments on articles which are being targeted for abuse. Problems with commenting? firstname.lastname@example.org