Moiré superlattices based on two-dimensional transition metal dichalcogenides (TMDs) have emerged as a highly versatile and fruitful platform for exploring correlated topological electronic phases. One of the most remarkable examples is the recently discovered fractional quantum anomalous Hall effect (FQAHE) under zero magnetic field. Here, we propose a minimal structure that hosts long-lived excitons-a ubiquitous bosonic excitation in TMD semiconductors-with narrow topological bosonic bands. The nontrivial exciton topology originates from hybridization of moiré interlayer excitons and is tunable by controlling twist angle and electric field. At small twist angle, the lowest exciton bands are isolated from higher energy bands and provide a solid-state realization of the bosonic Kane-Mele model with topological flatbands, which could potentially support the bosonic version of FQAHE.