We report the first self-catalyzed growth of high-quality GaSb nanowires on InAs stems using metal-organic chemical vapor deposition (MOCVD) on Si (111) substrates. To achieve the growth of vertical InAs/GaSb heterostructure nanowires, the two-step flow rates of the trimethylgallium (TMGa) and trimethylantimony (TMSb) are used. We first use relatively low TMGa and TMSb flow rates to preserve the Ga droplets on the thin InAs stems. Then, the flow rates of TMGa and TMSb are increased to enhance the axial growth rate. Because of the slower radial growth rate of GaSb at higher growth temperature, GaSb nanowires grown at 500 °C exhibit larger diameters than those grown at 520 °C. However, with respect to the axial growth, due to the Gibbs-Thomson effect and the reduction in the droplet supersaturation with increasing growth temperature, GaSb nanowires grown at 500 °C are longer than those grown at 520 °C. Detailed transmission electron microscopy (TEM) analyses reveal that the GaSb nanowires have a perfect zinc-blende (ZB) crystal structure. The growth method presented here may be suitable for other antimonide nanowire growth, and the axial InAs/GaSb heterostructure nanowires may have strong potential for use in the fabrication of novel nanowire-based devices and in the study of fundamental quantum physics.Electronic supplementary materialThe online version of this article (doi:10.1186/s11671-017-2207-5) contains supplementary material, which is available to authorized users.