The level structure of the neutron-rich ^{77}Cu nucleus is investigated through β-delayed γ-ray spectroscopy at the Radioactive Isotope Beam Factory of the RIKEN Nishina Center. Ions of ^{77}Ni are produced by in-flight fission, separated and identified in the BigRIPS fragment separator, and implanted in the WAS3ABi silicon detector array, surrounded by Ge cluster detectors of the EURICA array. A large number of excited states in ^{77}Cu are identified for the first time by correlating γ rays with the β decay of ^{77}Ni, and a level scheme is constructed by utilizing their coincidence relationships. The good agreement between large-scale Monte Carlo shell model calculations and experimental results allows for the evaluation of the single-particle structure near ^{78}Ni and suggests a single-particle nature for both the 5/2_{1}^{-} and 3/2_{1}^{-} states in ^{77}Cu, leading to doubly magic ^{78}Ni.