We formulate the Jost function formalism based on the Hartree-Fock-Bogoliubov (HFB) theory which has been used to represent the nature of nuclear superfluidity. By using the Jost function based on the HFB, the S-matrix for the neutron elastic scattering by the open-shell nucleus can be represented analytically. Adopting the Woods-Saxon potential, we calculate the S-matrix poles and their trajectories by varying the pairing strength using the Jost function in order to see the pairing effect on the complex energy-momentum plane. The total cross sections of neutron elastic scattering are also analyzed, and some sharp or asymmetrical resonances (from the configuration mixing) due to the pairing effect are found in the cross section.