The histone chaperone FACT occupies transcribed regions where it plays prominent roles in maintaining chromatin integrity and preserving epigenetic information. How it is targeted to transcribed regions, however, remains unclear. Proposed models for how FACT finds its way to transcriptionally active chromatin include docking on the RNA polymerase II (RNAPII) C-terminal domain (CTD), recruitment by elongation factors, recognition of modified histone tails and binding partially disassembled nucleosomes. Here, we systematically tested these and other scenarios in Saccharomyces cerevisiae and found that FACT binds transcribed chromatin, not RNAPII. Through a combination of experimental and mathematical modeling evidence, we propose that FACT recognizes the +1 nucleosome, as it is partially unwrapped by the engaging RNAPII, and spreads to downstream nucleosomes aided by the chromatin remodeler Chd1. Our work clarifies how FACT interacts with genes, suggests a processive mechanism for FACT function, and provides a framework to further dissect the molecular mechanisms of transcription-coupled histone chaperoning.