A numerical approach in scattering theory based on the path-integration technique is considered. An evaluation of a discretized Feynman path integral with an accurate Gaussian expansion of the integrand was used for the calculation of the differential cross sections for the scattering of a wave packet by a three-dimensional spherically symmetric Gaussian potential. The results are compared with the precise calculations using the phase functions method. This reveals separately discretization errors of the path integral, which depends on the single parameter---the number of discrete path points. High accuracy of the approach is demonstrated.