The structural phase transition and elastic properties of CoN are investigated by ab initio plane-wave pseudopotential density function theory method. The equilibrium lattice parameters a0, elastic constants Cij, bulk modulus B0 and its derivative B0 are calculated. From the usual condition of equal enthalpy, the phase transition of CoN from zinc-blende to rocksalt structure occurs at 35.4 GPa with a volume collapse of about 15.6%, consistent with the calculated result 36 GPa (FP-LDA), but an uncertainty is about 4.4 GPa compared with the 31 GPa (ASA-GGA). All three independent elastic constants, C11, C12, and C44 for CoN are calculated from direct computation of stresses generated by small strains. Both C12 and C44 are less sensitive to pressure as compared with C11. The calculated conclusions offer theoretical data for the further research of the mechanical properties for CoN.