The Italian - who also notched in the quarter-final victory over Chelsea - kept his composure in front of an expectant Stadium of Light crowd to ensure Gus Poyet's men travel to Old Trafford for the second leg with a slender advantage while also piling further misery on under-pressure United manager David Moyes.
In a tepid opening period, it was Sunderland who managed to find the breakthrough in first-half injury time when Wes Brown knocked Sebastian Larsson's free-kick across the six-yard box and Ryan Giggs inadvertently applied the final touch from close range.
United responded shortly after the interval when Nemanja Vidic, who has been linked with a move away from the Premier League champions, headed home a corner from the impressive Adnan Januzaj.
But, after substitute Adam Johnson was brought down in the box by Tom Cleverley on 65 minutes, Liverpool loanee Borini kept his cool to fire past David de Ge and leave the tie delicately poised ahead of the return fixture on January 22.
Both teams made seven changes, with Marcos Alonso making his debut for Sunderland while United named a strong line-up that included Patrice Evra, Michael Carrick and Vidic.
The hosts enjoyed the majority of possession in the opening exchanges and had the first glimpse of goal on 12 minutes with Borini's scuffed effort.
United began to settle into the match with Januzaj looking particularly threatening, and Moyes’ side went agonisingly close to breaking the deadlock when Giggs’ deflected strike rattled the crossbar after 25 minutes.
The visitors also thought their pressure had told on 38 minutes when Januzaj fired home after initially striking against Giggs, but the linesman ruled the latter was offside when he blocked the ball.
And United’s failure to take their chances counted against them on the stroke of half-time as Brown's cut-back across goal found its way home via Giggs, who was sliding in to challenge former United team-mate Phil Bardsley.
It did not take long for United to restore parity after the break, however, as Vidic leapt highest to head home Januzaj's whipped corner on 51 minutes.
Sunderland came close to regaining the lead on 62 minutes when Larsson's volley was parried to safety by De Gea.
Three minutes later, Poyet's side did move ahead when Johnson - who had replaced Emanuele Giaccherini from the bench - was felled by Cleverley in the area and Borini blasted the resulting penalty into the roof of the net.
With 20 minutes remaining, the lively Januzaj almost levelled with a bending effort that curled wide of Mannone's far post.
United piled on the pressure in the closing stages with Januzaj twice coming close and Evra's effort just evading the far post, but Sunderland withstood the barrage to ensure they will take a lead to Old Trafford.
