We report the numerical realization of robust two-component structures in 2D and 3D Bose-Einstein condensates with nontrivial topological charge in one component. We identify a stable symbiotic state in which a higher-dimensional bright soliton exists even in a homogeneous setting with defocusing interactions, due to the effective potential created by a stable vortex in the other component. The resulting vortex–bright-solitons, generalizations of the recently experimentally observed dark-bright solitons, are found to be very robust both in the homogeneous medium and in the presence of external confinement.