We present first-principles studies on how to engineer band lineups of nanosized radial heterojunctions, Si/Ge core-shell nanowires. Our calculation reveals that band offsets of these one-dimensional nanostructures can be tailored by applying the axial strain. In particular, the valence band offset can be efficiently tuned in a wide range and even be diminished with applied strain. Two mechanisms contributing to this strain engineering of band offsets are discussed. Our proposed approach to control band offsets in nanosized heterojunctions may be of practical interest for nanoelectronics and photovoltaic applications.