Topological aspects of the geometry of DNA and similar chiral molecules have received a lot of attention, but the topology of their electronic structure is less explored. Previous experiments revealed that DNA can efficiently filter spin-polarized electrons between metal contacts, a process called chiral-induced spin selectivity. However, the underlying correlation between chiral structure and electronic spin remains elusive. In this work, we reveal an orbital texture in the band structure, a topological characteristic induced by the chirality. We found that this orbital texture enables the chiral molecule to polarize the quantum orbital. This orbital polarization effect (OPE) induces spin polarization assisted by the spin-orbit interaction of a metal contact and leads to magnetoresistance and chiral separation. The orbital angular momentum of photoelectrons also plays an essential role in related photoemission experiments. Beyond chiral-induced spin selectivity, we predict that the orbital polarization effect could induce spin-selective phenomena even in achiral but inversion-breaking materials.