A quantum anomalous Hall system is predicted to realize topological superconductivity when proximity-coupled to an $s$-wave superconductor. A kagome magnet with chiral magnetic ordering exhibits the quantum anomalous Hall effect; however, superconducting proximity to an ordinary $s$-wave superconductor fails to induce pairing in the strong exchange coupling limit. In this work, we demonstrate that proximity coupling to a Rashba superconductor gives rise to topological superconducting phases characterized by odd Bogoliubov-de Gennes Chern numbers. We confirmed their consistency with the chiral central charge calculated based on the modular commutator. We also show that the magnetic ordering of kagome magnets is affected energetically by the proximity effect.