Paul Jewell's wait for a first victory as Derby manager ended at Sheffield Wednesday, where the Rams drew 1-1 and then won 4-2 on penalties.
Goalkeeper Lewis Price produced two superb penalty saves to help Jewell's beleaguered side progress to the FA Cup fourth round at the expense of Wednesday at Hillsborough.
After Kenny Miller had cancelled out Steve Watson's opener this third-round replay was settled on penalties.
Price saved from Glenn Whelan and Lunt while Miller, Lewis and Robert Earnshaw put Derby 3-1 up as Tudgay notched for Wednesday.
Burton made it 3-2 and it was left to Fagan to slot home the decisive spot-kick and set up a fourth-round home tie against Preston.