In this paper, we present a novel tracking controller for a class of uncertain nonaffine systems with time-varying asymmetric output constraints. Firstly, the original nonaffine constrained (in the sense of the output signal) control system is transformed into a output-feedback control problem of an unconstrained affine system in normal form. As a result, stabilization of the transformed system is sufficient to ensure constraint satisfaction. It is subsequently shown that the output tracking is achieved without violation of the predefined asymmetric time-varying output constraints. Therefore, we are capable of quantifying the system performance bounds as functions of time on both transient and steady-state stages. Furthermore, the transformed system is linear with respect to a new input signal and the traditional backstepping scheme is avoided, which makes the synthesis extremely simplified. All the signals in the closed-loop system are proved to be semi-globally, uniformly, and ultimately bounded via Lyapunov synthesis. Finally, the simulation results are presented to illustrate the performance of the proposed controller.