Genome engineering nucleases, including CRISPR-Cas12a, must access chromatinized DNA. Here, we investigate how Acidaminococcus sp. Cas12a cleaves DNA within human nucleosomes and phase-condensed nucleosome arrays. Using quantitative kinetics approaches, we show that dynamic nucleosome unwrapping regulates DNA target accessibility to Cas12a. Nucleosome unwrapping determines the extent to which both steps of Cas12a binding-PAM recognition and R-loop formation-are inhibited by the nucleosome. Nucleosomes inhibit Cas12a binding even beyond the canonical core particle. Relaxing DNA wrapping within the nucleosome by reducing DNA bendability, adding histone modifications, or introducing a target-proximal nuclease-inactive Cas9 enhances DNA cleavage rates over 10-fold. Surprisingly, Cas12a readily cleaves DNA linking nucleosomes within chromatin-like phase separated nucleosome arrays-with DNA targeting reduced only ~4-fold. This work provides a mechanism for the observation that on-target cleavage within nucleosomes occurs less often than off-target cleavage within nucleosome-depleted regions of cells. We conclude that nucleosome wrapping restricts accessibility to CRISPR-Cas nucleases and anticipate that increasing nucleosome breathing dynamics will improve DNA binding and cleavage in eukaryotic cells.