Goals from Cameron Jerome and Robert Huth ensured last season's FA Cup finalists Stoke City safely negotiated their way into the fifth round.
Striker Jerome's close-range effort in the fifth minute gave the visitors the lead after a nervy start from Derby.
The Championship side did enjoy periods of possession but struggled to test Stoke goalkeeper Asmir Begovic.
And in the 81st minute, Huth volley through a posse of players to deceive Begovic's counterpart Frank Fielding.
Derby had won their previous five home league and cup matches, but that form deserted them against a side eighth in the top-flight and preparing to play Valencia next month in the Europa League.
Nigel Clough's side went behind when they were undone by a short corner involving Matthew Etherington and Glenn Whelan.
Whelan delivered a low ball into the heart of the area, which was touched on by Marc Wilson before hitting Huth.
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