In this paper, the superconvergence error estimate of a low-order conforming mixed finite element scheme, which is called bilinear-constant scheme, for the Stokes equations with damping is established. In terms of the integral identity technique and dealing with the damping term carefully, the superclose estimates between the interpolation of the exact solution and the finite element solution for the velocity in $H^1$-norm and the pressure in $L^2$-norm are first derived. Then, the global superconvergence results for the velocity in $H^1$-norm and the pressure in $L^2$-norm are derived by a simple postprocessing technique with an economical workload. Finally, some numerical results are presented to demonstrate the correctness of the theoretical analysis.