Heterostructures of two-dimensional (2D) layered materials have attracted growing interest due to their unique properties and possible applications in electronics, photonics, and energy. Reduction of the dimensionality from 2D to one-dimensional (1D), such as graphene nanoribbons (GNRs), is also interesting due to the electron confinement effect and unique edge effects. Here, we demonstrate a bottom-up approach to grow vertical heterostructures of MoS2 and GNRs by a two-step chemical vapor deposition (CVD) method. Single-layer GNRs were first grown by ambient pressure CVD on an epitaxial Cu(100) film, followed by the second CVD process to grow MoS2 over the GNRs. The MoS2 layer was found to grow preferentially on the GNR surface, while the coverage could be further tuned by adjusting the growth conditions. The MoS2/GNR nanostructures show clear photosensitivity to visible light with an optical response much higher than that of a 2D MoS2/graphene heterostructure. The ability to grow a novel 1D heterostructure of layered materials by a bottom-up CVD approach will open up a new avenue to expand the dimensionality of the material synthesis and applications.