We study scattering of an electron by a screened potential of a nucleus in the field of a pulsed laser wave at small scattering angles. The interaction of an electron with the field of a nucleus is considered in the first Born approximation. An external field of a pulsed laser is accounted accurately as a quasimonochromatic wave. Analytical expressions are obtained for the transition amplitude and the cross section of the considered process. Scattering kinematics is defined at the minimal value of a transferred momentum. In this case the cross section contains a peak near the preferred scattering direction. It is shown that the maximum value of the cross section is determined by both the initial-electron energy and the energy of an external-field photon. Thus, the cross section of electron-nucleus scattering in a pulsed laser field can exceed in two orders of magnitude the cross section in absence of an external field in the case of ultrarelativistic energies and external field of a free-electron laser with keV-order photon energy.