A detailed theoretical study of the structural, elastic and vibrational properties of DyX (X = P, As) compounds is presented in this paper by performing ab initio calculations based on density functional theory using the VASP code. For describing the interaction between electrons and ions, the projector-augmented wave method is used. The generalized-gradient approximation is chosen for the exchange-correlation functional. The calculated structural parameters, such as the lattice constant, bulk modulus and second-order elastic constants, are presented. The high-pressure phase of both compounds is investigated and the phase transition pressure from NaCl (B1) to high-pressure phase (B2) is determined. To gain further information, we have calculated the Zener anisotropy factor (A), Poisson's ratio (nu), Young's modulus (E), shear modulus (C'), elastic wave velocities, Debye temperature, phonon frequencies and one-phonon density of states for the B1 structure of these compounds. The temperature-dependent variations of some thermodynamic properties such as entropy, heat capacity, internal energy and free energy are also predicted for the same compounds in the B1 phase.