Biometric authentication is the recognition of human identity via unique anatomical features. The development of novel methods parallels widespread application by consumer devices, law enforcement, and access control. In particular, methods based on finger veins, as compared to face and fingerprints, obviate privacy concerns and degradation due to wear, age, and obscuration. However, they are two-dimensional (2D) and are fundamentally limited by conventional imaging and tissue-light scattering. In this work, for the first time, to the best of our knowledge, we demonstrate a method of three-dimensional (3D) finger vein biometric authentication based on photoacoustic tomography. Using a compact photoacoustic tomography setup and a novel recognition algorithm, the advantages of 3D are demonstrated via biometric authentication of index finger vessels with false acceptance, false rejection, and equal error rates <1.23%, <9.27%, and <0.13%, respectively, when comparing one finger, a false acceptance rate improvement >10× when comparing multiple fingers, and <0.7% when rotating fingers ±30.