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Moir\'e superlattices of layered transition metal dichalcogenides are proven to host periodic electron crystals due to strong correlation effects. These electron crystals can also be intertwined with intricate magnetic phenomena. In this Letter, we present our findings on the moir\'e exchange effect, resulting from the modulation of local magnetic moments by electron crystals within well-aligned WSe₂/WS₂ heterobilayers. Employing polarization-resolved magneto-optical spectroscopy, we unveil a high-energy excitonic resonance near one hole per moir\'e unit cell (v=-1), which possesses a giant g factor several times greater than the already very large g factor of the WSe₂ A exciton in this heterostructure. Supported by continuum model calculations, these high-energy states are found to be dark excitons brightened through Umklapp scattering from the moir\'e mini-Brillouin zone. When the carriers form a Mott insulating state near v=-1, the Coulomb exchange between doped carriers and excitons forms an effective magnetic field with moir\'e periodicity. This moir\'e exchange effect gives rise to the observed giant g factor for the excitonic Umklapp state.
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