Using first-principles density functional calculations, a systematic study on the elastic properties for all known superconducting MAX phases (Nb2SC, Nb2SnC, Nb2AsC, Nb2InC, Mo2GaC and Ti2InC) was performed. As a result, the optimized lattice parameters, independent elastic constants, indicators of elastic anisotropy and brittle/ductile behavior as well as the so-called machinability indexis were calculated. We derived also bulk and shear moduli, Young's moduli, and Poisson's ratio for ideal polycrystalline MAX aggregates. The results obtained were discussed in comparison with available theoretical and experimental data and elastic parameters for other layered superconductors.