We propose a feasible way of engineering Majorana-type quasiparticles in ultracold fermionic gases on a one-dimensional (1D) optical lattice. For this purpose, imbalanced ultracold atoms interacting by the spin-orbit coupling should be hybridized with a three-dimensional Bose-Einstein condensate molecular cloud. We show that the Majorana-type excitations can be created or annihilated upon constraining the profile of a trapping potential and/or an internal scattering barier. This process is modeled within the Bogoliubov-de Gennes approach. Our study is relevant also to nanoscopic 1D superconductors, where both potentials can be imposed by electrostatic means.