Moiré lattices created in two-dimensional heterostructures exhibit rich many-body physics of interacting electrons and excitons and, at the same time, suggest promising optoelectronic applications. Here, we study the cooperative radiance of moiré excitons that is demonstrated to emerge from the deep subwavelength nature of the moiré lattice and the strong excitonic on-site interaction. In particular, we show that the static dipole-dipole interaction between interlayer excitons can strongly affect their cooperative optical properties, suppressing superradiance of disordered states while enhancing superradiance of ordered phases of moiré excitons. Moreover, we show that doping permits direct control of optical cooperativity, e.g., by generating superradiant dynamics of otherwise subradiant states of excitons. Our results show that interlayer moiré excitons offer a unique platform for exploring cooperative optical phenomena in strongly interacting many-body systems, thus holding promise for applications in quantum nonlinear optics.