To elucidate how eukaryotic sequence-specific transcription factors (TFs) search for gene targets on chromatin, we used multi-color smFRET and single-particle imaging to track the diffusion of purified GAGA-Associated Factor (GAF) on DNA and nucleosomes. Monomeric GAF DNA-binding domain (DBD) bearing one zinc finger finds its cognate site by 1D or 3D diffusion on bare DNA and rapidly slides back-and-forth between naturally clustered motifs for seconds before escape. Multimeric, full-length GAF also finds clustered motifs on DNA by 1D-3D diffusion, but remains locked on target for longer periods. Nucleosome architecture effectively blocks GAF-DBD 1D-sliding into the histone core but favors retention of GAF-DBD when targeting solvent-exposed sites by 3D-diffusion. Despite the occlusive power of nucleosomes, 1D-3D facilitated diffusion enables GAF to effectively search for clustered cognate motifs in chromatin, providing a mechanism for navigation to nucleosome and nucleosome-free sites by a member of the largest TF family.