FeTe has long been regarded as a nonsuperconducting antiferromagnetic metal with trivial band topology, but recent advances in stoichiometry control have begun to challenge this picture. Here we use higher-order epitaxy on zinc-blende MnTe to stabilize near-stoichiometric FeTe with strongly suppressed interstitial Fe and a superconducting transition onset near 13 K. Angle-resolved photoemission spectroscopy reveals markedly enhanced quasiparticle coherence, well-defined Fe-derived hole bands, and a nearly two-dimensional Dirac-cone-like state near the Fermi level. First-principles calculations identify an inversion between odd- and even-parity bands, yielding nontrivial Z₂ topology and a Dirac surface state consistent with experiment. These results elucidate the intrinsic electronic structure of stoichiometric superconducting FeTe and provide evidence for nontrivial band topology, positioning FeTe/MnTe as a promising platform for exploring topological superconductivity.