Spectroscopic properties that characterize shape-phase transitions in neutron-rich odd-A Zr isotopes are investigated using the framework of nuclear density-functional theory and particle-core coupling. The interacting-boson Hamiltonian of the even-even core nuclei, and the single-particle energies and occupation probabilities of the unpaired neutron, are completely determined by deformation constrained self-consistent mean-field calculations based on the relativistic Hartree-Bogoliubov model with a choice of a universal energy density functional and pairing interaction. The triaxial (β,γ ) deformation energy surfaces for even-even 94−102Zr indicate the occurrence of a transition from triaxial or γ soft (94,96Zr) to prolate (98Zr) and triaxial (100,102Zr) shapes. The corresponding low-energy excitation spectra of the odd-A Zr isotopes are in very good agreement with recent experimental results. Consistent with the structural evolution of the neighboring even-even Zr nuclei, the state-dependent effective deformations and their fluctuations in the odd-A isotopes indicate a pronounced discontinuity around the transitional nucleus 99Zr.