Controlling matter-light interactions with cavities is of fundamental importance in modern science and technology(1). This is exemplified in the strong-coupling regime, where matter-light hybrid modes form, with properties that are controllable by optical-wavelength photons(2,3). By contrast, matter excitations on the nanometre scale are harder to access. In two-dimensional van der Waals heterostructures, a tunable moire lattice potential for electronic excitations may form(4), enabling the generation of correlated electron gases in the lattice potentials(5-9). Excitons confined in moire lattices have also been reported(10,11), but no cooperative effects have been observed and interactions with light have remained perturbative(12-15). Here, by integrating MoSe2-WS2 heterobilayers in a microcavity, we establish cooperative coupling between moire-lattice excitons and microcavity photons up to the temperature of liquid nitrogen, thereby integrating versatile control of both matter and light into one platform. The density dependence of the moire polaritons reveals strong nonlinearity due to exciton blockade, suppressed exciton energy shift and suppressed excitation-induced dephasing, all of which are consistent with the quantum confined nature of the moire excitons. Such a moire polariton system combines strong nonlinearity and microscopic-scale tuning of matter excitations using cavity engineering and long-range light coherence, providing a platform with which to study collective phenomena from tunable arrays of quantum emitters.