The integration of different two-dimensional materials within a multilayer van der Waals (vdW) heterostructure offers a promising technology for high performance opto-electronic devices such as photodetectors and light sources. Here we report on the fabrication and electronic properties of vdW heterojunction diodes composed of the direct band gap layered semiconductors InSe and GaSe and transparent monolayer graphene electrodes. We show that the type II band alignment between the two layered materials and their distinctive spectral response, combined with the short channel length and low electrical resistance of graphene electrodes, enable efficient generation and extraction of photoexcited carriers from the heterostructure even when no external voltage is applied. Our devices are fast (∼2 μs), self-driven photodetectors with multicolor photoresponse ranging from the ultraviolet to the near-infrared and offer new routes to miniaturized optoelectronics beyond present semiconductor materials and technologies.