Terahertz (THz) photoconductive devices are used for generation, detection and modulation of THz waves, and they rely on the ability to switch electrical conductivity on a subpicosecond time scale using optical pulses. However, fast and efficient conductivity switching with high contrast has been a challenge, because the majority of photoexcited charge carriers in the switch do not contribute to the photocurrent due to fast recombination. Here, we improve efficiency of electrical conductivity switching using a network of electrically connected nano-scale GaAs resonators, which form a perfectly-absorbing photoconductive metasurface. We achieve perfect absorption without incorporating metallic elements, by breaking the symmetry of cubic Mie resonators. As a result, the metasurface can be switched between conductive and resistive states with extremely high contrast using an unprecedentedly low level of optical excitation. We integrate this metasurface with a THz antenna to produce an efficient photoconductive THz detector. The perfectly-absorbing photoconductive metasurface opens paths for developing a wide range of efficient optoelectronic devices, where required optical and electronic properties are achieved through nanostructuring the resonator network.