A linear mechanical system with constant matrix of dissipative forces and continuous and bounded matrices of positional forces is studied. It is assumed that there are a large positive multiplier at the vector of dissipative forces and a constant delay in positional forces. The case is considered where the associated delay-free averaged system is asymptotically stable. With the aid of the Lyapunov direct method and the averaging method, conditions are derived under which neither delay nor timevarying perturbations with zero mean values disturb the asymptotic stability. The developed approach is used in the problem of monoaxial stabilization of a rigid body.