In this paper, a numerical study is made of an initial-boundary value problem for a singularly perturbed delay Sobolev equations (SPDSEs). Here we propose an exponentially fitted method based on finite differences to solve an SPDSE. An exponentially-fitted difference scheme on a uniform mesh, which is accomplished by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form, is presented. We calculate the fitting parameter for an exponentially fitted finite difference scheme corresponding to the problem and establish an error estimate which shows that the method has order of convergence 2 in space and time, independently of the perturbation parameter to the solution of the problem. The stability of the method is discussed. Numerical experiments are performed to support the theoretical results.