The Fermi Bubbles are giant, γ-ray emitting lobes emanating from the nucleus of the Milky Way[1, 2] discovered in ∼ 1-100 GeV data collected by the Large Area Telescope on board the Fermi Gamma-Ray Space Telescope[3]. Previous work[4] has revealed substructure within the Fermi Bubbles that has been interpreted as a signature of collimated outflows from the Galaxy's super-massive black hole. Here we show that much of the γ-ray emission associated to the brightest region of substructure -the so-called cocoon -is actually due to the Sagittarius dwarf spheroidal (Sgr dSph) galaxy. This large Milky Way satellite is viewed through the Fermi Bubbles from the position of the Solar System. As a tidally and ram-pressure stripped remnant, the Sgr dSph has no on-going star formation, but we demonstrate that its γ-ray signal is naturally explained by inverse Compton scattering of cosmic microwave background photons by high-energy electron-positron pairs injected by the dwarf's millisecond pulsar (MSP) population, combined with these objects' magnetospheric emission. This finding suggests that MSPs likely produce significant γ-ray emission amongst old stellar populations, potentially confounding indirect dark matter searches in regions such as the Galactic Centre, the Andromeda galaxy, and other massive Milky Way dwarf spheroidals.Early analysis of data from the Fermi Large Area Telescope (Fermi-LAT) identified two counter-propagating, co-linear γ-ray substructures within the Fermi Bubbles (FBs; Figure 1a), a jet in the northern Galactic hemisphere and cocoon in the south[4]; subsequent, independent analyses[2, 5] have only confirmed the existence of the latter. Since the cocoon is contained within the solid angle of the surrounding FBs and exhibits a similar γ-ray spectrum, it is natural to propose they share a common