Quasicrystals are emerging topological materials which defy all the laws of crystallography, with aperiodic long-range order and peculiar rotational symmetry associated with higher-dimensional space group. We study topological superconductivity with broken time-reversal symmetry in prototypical two-dimensional quasicrystals, the Penrose and the Ammann-Beenker quasicrystals. By solving the Bogoliubov-de Gennes equations self-consistently for the superconducting mean fields, we show the stable occurrence of topological superconductivity in both Penrose and Ammann-Beenker quasicrystals. The topological nature of superconductivity in a quasicrystal is signified by the Bott index, which is equivalent to the first Chern number in periodic systems. We confirm the appearance of a zero-energy Majorana fermion at a surface boundary or in a vortex when the Bott index is unity, in accordance with the bulk-boundary correspondence and the non-Abelian character of the system.