This paper studies distributed singularity-free finite-time consensus tracking control for quite a large class of high-order (powers are positive odd integers) nonlinear multi-agent networks in the presence of unknown asymmetric dead zone. Achieving finite-time consensus tracking for such dynamics is extremely challenging because feedback linearization and backstepping methods successfully developed for low-order systems fail to work, and some appropriate exponential terms typically arising from finite-time stability are difficult to design due to the existence of high powers and strong couplings among distinct agents. To this purpose, an adding-one-power-integrator methodology is skillfully incorporated into the finite-time stability theory so as to stabilize the closed-loop system. Over the course of design, a variable-separable lemma is utilized to extract the unknown asymmetric dead-zone input in a “linear-like” manner and fuzzy logic systems are utilized to estimate the unknown system continuous nonlinearities over some compact sets. The singularity issue typical of finite-time control is overcome by delicately introducing a switching function. It is rigorously proved that the consensus tracking error eventually converges to a residual set, whose size can be made as small as desired, in finite time, while guaranteeing the boundedness of all closed-loop signals. Comparative simulations are finally provided to verify the effectiveness of the presented scheme on the existing control methods.