This paper presents output tracking control for strict feedback nonlinear systems with time-varying output constraint. A Barrier Lyapunov Function (BLF), which depends explicitly on time, is employed at the outset to prevent the output from violating the time-varying constraint. Specifically, we allow the barrier limit to vary with the desired trajectory in time. Through a change of coordinates for the tracking error, we then eliminate the time dependence, therefore simplifying the analysis. We show that asymptotic output tracking is achieved without violation of the time-varying constraint, and that all closed loop signals remain bounded. The performance of the proposed control is illustrated through a simulation example.