The B1 (NaCl) and B2 (CsCl) structure of rare-earth monochalcogenides XY (X = La, Ce, Eu, and Y = S, Se, and Te) were investigated with the full-potential linearized-augmented plane wave (FP-LAPW) scheme in the frame of the generalized gradient approximation (GGA). The optimized lattice parameters, independent elastic constants (C11, C12, and C44) at zero and high pressure, bulk modulus B, and its pressure derivative and the shear modulus G were evaluated. Further, the numerical estimates of a set of elastic parameters [Young's modulus E, Poisson's ratio (ν), Lame's coefficients (µ, λ)] of the polycrystalline XY (X = La, Ce, Eu, and Y = S, Se, and Te) compounds (in the framework of the Voigt–Reuss–Hill approximation) were performed. The pressures at which these compounds undergo structural phase transition from B1 (NaCl) to B2 (CsCl) phases were calculated. For rare-earth monochalcogenides XY, the Debye temperature is also estimated from the average sound velocity.