The Razumikhin-type approach is introduced to solve the output-feedback stabilization problem for a class of nonlinear systems with time-varying delay. By using the Razumikhin-type theorem and the backstepping design method, effective observer and suitable Lyapunov function, an output-feedback controller is constructed to ensure the origin of closed-loop system globally asymptotically stable. The limitations on the derivative of delay can be completely removed, which is the common assumption of nonlinear systems with time-varying delay. The efficiency of the output-feedback controller is demonstrated by a simulation example.