© 2017 The Authors. Among the most important goals in cosmology is detecting and quantifying small (Mhalo ≃ 106-9 M⊙) dark matter (DM) subhaloes. Current probes around the Milky Way (MW) are most sensitive to such substructure within ~20 kpc of the halo centre, where the galaxy contributes significantly to the potential.We explore the effects of baryons on subhalo populations in ΛCDM using cosmological zoom-in baryonic simulations of MW-mass haloes from the Latte simulation suite, part of the Feedback In Realistic Environments (FIRE) project. Specifically, we compare simulations of the same two haloes run using (1) DM-only (DMO), (2) full baryonic physics and (3) DM with an embedded disc potential grown to match the FIRE simulation. Relative to baryonic simulations, DMO simulations contain ~2 × as many subhaloes within 100 kpc of the halo centre; this excess is ≳5 × within 25 kpc. At z = 0, the baryonic simulations are completely devoid of subhaloes down to 3 × 106M⊙ within 15 kpc of the MW-mass galaxy, and fewer than 20 surviving subhaloes have orbital pericentres < 20 kpc. Despite the complexities of baryonic physics, the simple addition of an embedded central disc potential to DMO simulations reproduces this subhalo depletion, including trends with radius, remarkably well. Thus, the additional tidal field from the central galaxy is the primary cause of subhalo depletion. Subhaloes on radial orbits that pass close to the central galaxy are preferentially destroyed, causing the surviving population to have tangentially biased orbits compared to DMO predictions. Our method of embedding a potential in DMO simulations provides a fast and accurate alternative to full baryonic simulations, thus enabling suites of cosmological simulations that can provide accurate and statistical predictions of substructure populations.