The recently emerged organohalide perovskites (e.g., CH3NH3PbI3) have drawn intense attention for high efficiency solar cells. However, with a considerable solubility in many solvents, these perovskites are not typically compatible with conventional lithography processes for more complicated device fabrications that are important for both fundamental studies and technological applications. Here we report the creation of novel heterojunction devices based on perovskites and two-dimensional (2D) crystals by taking advantage of the layered characteristic of lead iodide (PbI2) and vapor phase intercalation. We show a graphene/perovskite/graphene vertical stack can deliver a highest photoresponsivity of ~950 A/W and photoconductive gain of ~2200, and a graphene/WSe2/perovskite/graphene heterojunction can display a high on/off ratio (~10^6) transistor behavior with distinct gate-tunable diode characteristics and open-circuit voltages. Such unique perovskite-2D heterostructures have significant potential for future optoelectronic research and can enable broad possibilities with compositional tunability of organohalide perovskites and the versatility offered by diverse 2D materials.