Moiré materials have emerged as a powerful platform for exploring exotic quantum phases. While recent experiments have unveiled fractional Chern insulators exhibiting the fractional quantum anomalous Hall effect based on electrons or holes, the exploration of analogous many-body states with bosonic constituents remains largely uncharted. In this work, we predict the emergence of bosonic fractional Chern insulators arising from long-lived excitons in a moiré superlattice formed by twisted bilayer WSe₂ stacked on monolayer MoSe₂. Performing exact diagonalization on the exciton flat Chern band present in this structure, we provide compelling evidence for the existence of Abelian and non-Abelian phases at band filling and 1, respectively, through multiple robust signatures, including ground-state degeneracy, spectral flow, many-body Chern number, and particle-cut entanglement spectrum. The obtained energy gap of ∼10 meV for the Abelian states suggests a remarkably high stability of this phase, which persists for a relatively wide range of twist angles and vertical electric fields. Our findings establish the presence of robust bosonic fractional Chern insulators in highly tunable and experimentally accessible moiré heterostructures and unveil a promising pathway for realizing non-Abelian anyons.