After Kenny Miller had cancelled out Steve Watson’s opener, this third-round replay was settled on penalties.
Price was the hero of the hour as he saved from Glenn Whelan and Kenny Lunt to give Rams manager Paul Jewell his first victory since succeeding Billy Davies in November.
After extra time failed to separate the sides, Price saved penalties from Glenn Whelan and Lunt while Miller, Lewis and Earnshaw put Derby 3-1 up and 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.
SHEFF WED: Grant, Bullen, Hinds, Beevers, Spurr, Whelan, Watson (Lunt 51), Johnson, Small (Burton 81), Sodje (Clarke 71), Tudgay.
DERBY: Price, Edworthy, Leacock (Moore 83), Todd, Davis, Nyatanga, Teale (Earnshaw 106), Pearson, Barnes (Lewis 42), Fagan, Miller.
* Meanwhile, after a 1-1 draw at Underhill, League Two Barnet earned a fourth-round tie against Bristol Rovers after seeing off League One Swindon Town 2-0 on penalties.