Carlos Tevez, who returned to United's starting line-up after a new trip to his native Argentina, was particulary ineffective and, for the most part, Derby looked comfortably with their lead.
United did improve before the end, with Cristiano Ronaldo dragging one shot wire before firing in a free-kick that fizzed just the wrong side of Carroll's post.
United will still be favourites to overturn their single-goal deficit in the second leg to Old Trafford on 20 January.
But a more immediate concern for Alex Ferguson is the weekend visit of Luiz Felipe Scolari's side, and he will know a significant improvement on his performance is needed.
United: Kuszczak, Evans, Vidic, O'Shea, R. Da Silva, Anderson (Carrick 74), Scholes (Ronaldo 63), Gibson, Nani, Tevez, Welbeck (Rooney 63).
Subs not used: Amos, Giggs, Fletcher, Possebon.
Booked: R. Da Silva.
Referee: Phil Dowd.
Man-of-the-Match: Kris Commons (Derby County).
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