We implemented a novel technique to perform the collective spectral analysis of sets of multiple gamma-ray point sources using the data collected by the Large Area Telescope onboard the Fermi satellite. The energy spectra of the sources are reconstructed starting from the photon counts and without assuming any spectral model for both the sources and the background. In case of faint sources, upper limits on their fluxes are evaluated with a Bayesian approach. This analysis technique is very useful when several sources with similar spectral features are studied, such as sources of gamma rays from annihilation of dark matter particles. We present the results obtained by applying this analysis to a sample of dwarf spheroidal galaxies and to the Milky Way dark matter halo. The analysis of dwarf spheroidal galaxies yields upper limits on the product of the dark matter pair annihilation cross section and the relative velocity of annihilating particles that are well below those predicted by the canonical thermal relic scenario in a mass range from a few GeV to a few tens of GeV for some annihilation channels. ; Comment: 17 pages; 11 figures