If dark matter is composed of neutralinos, one of the most exciting prospects for its detection lies in observations of the gamma-ray radiation created in pair annihilations between neutralinos, a process that may contribute significantly to the extragalactic gamma-ray background (EGB) radiation. We here use the high-resolution Millennium-II simulation of cosmic structure formation to produce the first full-sky maps of the expected radiation coming from extragalactic dark matter structures. Our map making procedure takes into account the total gamma-ray luminosity from all haloes and their subhaloes, and includes corrections for unresolved components of the emission as well as an extrapolation to the damping scale limit of neutralinos. Our analysis also includes a proper normalization of the signal according to a specific supersymmetric model based on minimal supergravity. The new simulated maps allow a study of the angular power spectrum of the gamma-ray background from dark matter annihilation, which has distinctive features associated with the nature of the annihilation process and may be detectable in forthcoming observations by the recently launched FERMI satellite. Our results are in broad agreement with analytic models for the gamma-ray background, but they also include higher-order correlations not readily accessible in analytic calculations and, in addition, provide detailed spectral information for each pixel. In particular, we find that difference maps at different energies can reveal cosmic large-scale structure at low and intermediate redshifts. If the intrinsic emission spectrum is characterized by an emission peak, cosmological tomography with gamma ray annihilation radiation is in principle possible.