The Liverpool loanee's second-half spot kick saw Gus Poyet's men pick up a crucial win at the Stadium of Light after Nemanja Vidic had cancelled out Ryan Giggs' own goal.
The visitors saw Ryan Giggs smash the crossbar from range while Adnan Januzaj had a goal ruled out for offside in the first half, but it was Gus Poyet's men who took the lead in stoppage time of the first period as United's veteran Welshman bundled the ball into his own net.
The Red Devils responded immediately after the restart, Nemanja Vidic levelling the score from a corner, but when Tom Cleverley felled substitute Adam Johnson inside the box, Liverpool loanee Borini was able to restore the hosts' lead from the penalty spot.
Both teams made seven changes, with Marcos Alonso making his debut for Sunderland, while United named a strong lineup 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 12 minutes in with Borini's scuffed effort.
United began to settle into the match with Januzaj looking particularly threatening, and Moyes’ side went agonizingly close to breaking the deadlock when Giggs’ deflected strike rattled the crossbar after 25 minutes.
The visitors also thought their pressure had paid off in the 38th minute 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 its chances counted against it on the stroke of halftime as Wes Brown knocked Sebastian Larsson's free kick across the 6-yard box where Giggs, sliding in to challenge former United teammate Phil Bardsley, inadvertently applied the final touch from close range.
It did not take long for United to restore parity after the break, however, as Vidic leapt highest to head home Cleverley's whipped corner in the 51st minute.
But, rather than United building on its equalizer, it was Sunderland that came closest to regaining the lead in the 62nd minute when Larsson's volley was parried to safety by David de Gea.
Three minutes later, Poyet's side did move ahead when Johnson - who had replaced Emanuele Giaccherini from the bench - was the victim of a clumsy tackle 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 Vito Mannone's far post. United then piled on the pressure in the closing stages, with Januzaj twice coming close and an effort from Evra just evading the far post. But Sunderland withstood the barrage to ensure it will take a lead to Old Trafford in the second leg Jan. 22.