We investigate the elastic properties of selected zinc-blende III-V semiconductors. Using hybrid functional density functional theory, we calculate the second- and third-order elastic constants and first- and second-order internal strain tensor components for Ga, In, and Al containing III-V compounds. For many of these parameters, there are no available experimental measurements, and this work is the first to predict their values. The stricter convergence criteria for the calculation of higher-order elastic constants are demonstrated, and arguments are made based on this for extracting these constants via the calculated stresses, rather than the energies, in the context of plane-wave-based calculations. The calculated elastic properties are used to determine the strain regime at which higher-order elasticity becomes important by comparing the stresses predicted by a lower- and a higher-order elasticity theory. Finally, the results are compared with available experimental literature data and previous theory.