Organic-inorganic perovskites have already shown a tremendous potential for low-cost light-harvesting devices. Yet, the relatively low carrier mobilities in bulk perovskites still prevent large-area devices with performances competing with state-of-the-art technologies. Here, we tackle this fundamental challenge by incorporating single-wall carbon nanotubes within a perovskite matrix by means of a simple two-step method in ambient air. Using this nano-engineered hybrid film, we demonstrate large-area photodetectors with responsivities up-to 13.8 A.W-1 and a broad spectral response from 300 to 800 nm, indicating that photocurrent generation arises from the charge transfer from the perovskite matrix to the embedded nanotube network. As the nanotubes facilitate the carrier extraction, these photodetectors also show a fast response time of 10 ms. This is significantly faster than most of previous reports on perovskite-based photodetectors, including devices with much smaller photosensitive areas. This approach is also well-suited for large-scale production of other perovskite-based light-harvesting devices.