Explicitly connected many-body perturbation expansion for the energy of the first-order exchange interaction between closed-shell atoms or molecules is derived. The influence of the intramonomer electron correlation is accounted for by a perturbation expansion in terms of the Moller-Plesset fluctuation potentials W-A and W-B of the monomers or by a nonperturbative coupled-cluster type procedure. Detailed orbital expressions for the intramonomer correlation corrections of the first and second order in W-A+W-B are given. Our method leads to novel expressions for the exchange energies in which the exchange and hybrid integrals do not appear. These expressions, involving only the Coulomb and overlap integrals, are structurally similar to the standard many-body perturbation theory expressions for the polarization energies. Thus, the exchange corrections can be easily coded by suitably modifying the existing induction and dispersion energy codes. As a test of our method we have performed calculations of the first-order exchange energy for the He-2, (H-2)(2), and He-H-2 complexes. The results of the perturbative calculations are compared with the full configuration interaction data computed using the same basis sets. It is shown that the Moller-Plesset expansion of the first-order exchange energy converges moderately fast, whereas the nonperturbative coupled-cluster type approximations reproduce the full configuration interaction results very accurately.