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We develop a systematic theory for excitons subject to Fermi-Hubbard physics in moiré twisted transition metal dichalcogenides (TMDs). Specifically, we consider excitons from two moiré bands with a Mott-insulating valence band sustaining 120 spin order. These “Mott-moiré excitons,” which are achievable in twisted TMD heterobilayers, are bound states of a magnetic polaron in the valence band and a free electron in the conduction band. We find significantly narrower exciton bandwidths in the presence of Hubbard physics, serving as a potential experimental signature of strong correlations. We also demonstrate the high tunability of Mott-moiré excitons through the dependence of their binding energies, diameters, and bandwidths on the moiré period. In addition, we study bound states between charges outside of the strongly correlated moiré band and find that these as well exhibit signatures of spin correlation. Our work provides guidelines for future exploration of strongly correlated excitons in triangular Hubbard systems such as twisted TMD heterobilayers.