The Black Cats claimed their first victory over United since November 2000 thanks to Fabio Borini's second-half penalty at the Stadium of Light as the Red Devils suffered a third consecutive defeat in all competitions, their worst run since May 2001.
Borini fired high into David de Gea's net in the 64th minute on Tuesday night after Tom Cleverley was penalised for a clumsy challenge on substitute Adam Johnson as he burst into the box.
Sunderland took the lead on the stroke of half-time as Ryan Giggs could only turn the ball into his own net under pressure from Phil Bardsley after Wes Brown's pass following Seb Larsson's free-kick.
United equalised early in the second period as Nemanja Vidic headed home Cleverley's corner, but Borini's penalty means that more questions will be asked about Moyes' management, even though there is still time for redemption in the second leg at Old Trafford on January 22.
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 bar.
With Carrick starting to pull the strings and Patrice Evra and 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 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.
Moyes' disappointment was to increase in injury time at the end of the first half when Brown turned Larsson's free-kick back across goal and Giggs bundled it under his own net under extreme pressure from Bardsley.
It took the visitors just seven minutes of the second half to restore parity, as Cleverley floated a 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 De Gea minutes earlier, refused to succumb and they regained the lead when Cleverley tripped Johnson, with referee Andre 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, Rafael was booked for protesting - and was then fortunate to stay on the field after catching Borini soon after.
Borini sent the keeper the wrong way 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.