We present the first detailed elemental abundances in the ultra-faint Magellanic satellite galaxies Carina II (Car II) and Carina III (Car III). With high-resolution Magellan/MIKE spectroscopy, we determined the abundances of nine stars in Car II, including the first abundances of an RR Lyrae star in an ultra-faint dwarf galaxy (UFD), and two stars in Car III. The chemical abundances demonstrate that both systems are clearly galaxies and not globular clusters. The stars in these galaxies mostly display abundance trends matching those of other similarly faint dwarf galaxies: enhanced but declining [α/Fe] ratios, iron-peak elements matching the stellar halo, and unusually low neutron-capture element abundances. One star displays a low outlying [Sc/Fe] = −1.0. We detect a large Ba scatter in Car II, likely due to inhomogeneous enrichment by low-mass asymptotic giant branch star winds. The most striking abundance trend is for [Mg/Ca] in Car II, which decreases from +0.4 to −0.4 and indicates clear variation in the initial progenitor masses of enriching core-collapse supernovae. So far, the only UFDs displaying a similar [Mg/Ca] trend are likely satellites of the Large Magellanic Cloud. We find two stars with [Fe/H] ≤ −3.5 whose abundances likely trace the first generation of metal-free Population III stars and are well fit by Population III core-collapse supernova yields. An appendix describes our new abundance uncertainty analysis that propagates line-by-line stellar parameter uncertainties.