We present a homogeneous chemical abundance analysis of five of the most metal-poor stars in the Sculptor dwarf spheroidal galaxy. We analyze new and archival high resolution spectroscopy from Magellan/MIKE and VLT/UVES and determine stellar parameters and abundances in a consistent way for each star. Two of the stars in our sample, at [Fe/H]=-3.5 and [Fe/H]=-3.8, are new discoveries from our Ca K survey of Sculptor, while the other three were known in the literature. We confirm that Scl 07-50 is the lowest metallicity star identified in an external galaxy, at [Fe/H]=-4.1. The two most metal-poor stars both have very unusual abundance patterns, with striking deficiencies of the {alpha} elements, while the other three stars resemble typical extremely metal-poor Milky Way halo stars. We show that the star-to-star scatter for several elements in Sculptor is larger than that for halo stars in the same metallicity range. This scatter and the uncommon abundance patterns of the lowest metallicity stars indicate that the oldest surviving Sculptor stars were enriched by a small number of earlier supernovae, perhaps weighted toward high-mass progenitors from the first generation of stars the galaxy formed.