Wayne Routledge's opener for the Swans was quickly cancelled out by Javier Hernández, but following substitute Fábio da Silva's red card Swansea struck the killer blow in the 90th minute.
It is only the third time in 30 years United have been eliminated from the Cup at the third round stage, and it is their fifth Old Trafford defeat under Moyes, four days on from Tottenham's victory in M16.
Bereft of Robin van Persie and Wayne Rooney, United looked a shadow of the determined champions under Sir Alex Ferguson last season during an uninspiring encounter.
Inevitably, Moyes turned to 18-year-old substitute Adnan Januzaj for inspiration, which highlights how incompetent the Old Trafford hierarchy were when they barely strengthened the squad last summer. However his task was made tougher when the over-eager Fábio was shown a straight red card for José Cañas.
Darren Fletcher switched to right-back and was easily outpaced by Routledge, whose cross was nodded in by the unmarked Bony.