Resolution of Holliday junctions is a critical intermediate step of homologous recombination in which junctions are processed by junction-resolving endonucleases. Although binding and cleavage are well understood, the question remains how the enzymes locate their substrate within long duplex DNA. Here we track fluorescent dimers of endonuclease I on DNA, presenting the complete single-molecule reaction trajectory for a junction-resolving enzyme finding and cleaving a Holliday junction. We show that the enzyme binds remotely to dsDNA and then undergoes 1D diffusion. Upon encountering a four-way junction, a catalytically-impaired mutant remains bound at that point. An active enzyme, however, cleaves the junction after a few seconds. Quantitative analysis provides a comprehensive description of the facilitated diffusion mechanism. We show that the eukaryotic junction-resolving enzyme GEN1 also undergoes facilitated diffusion on dsDNA until it becomes located at a junction, so that the general resolution trajectory is probably applicable to many junction resolving enzymes.