The Owls made a flying start and needed only 10 minutes to take the lead.
Steve Watson provided the finish after Jermaine Johnson was afforded too much time and space on the right wing and his cross was steered past keeper Lewis Price by the veteran midfielder from just inside the area.
Derby almost went two down on the half hour when centre-back Lewin Nyatanga was forced to clear off the line after Johnson caused more havoc.
But the Rams emerged fired-up after the break and Kenny Miller found space in the left-hand corner of the box and curled an exquisite shot around keeper Lee Grant.
Derby then shaded the rest of the encounter and profited when Kenny Lunt and Glenn Whelan missed in the shoot-out.