In this paper, the stabilising control problem for a class of nonlinear system in the presence of measurement delay is addressed. A full-order observer is designed to eliminate the effect of variable output time delay, which is bounded and known. Then, the estimated states are utilised for the state feedback control law to stabilise the considered control system. Lyapunov-Razumikhin approach is employed to analyse the stability of the closed-loop system. Unlike the previous work, the exponential convergence of the estimation error is ensured, rather than asymptotic convergence, by designing a delay-dependent gain of the observer. Moreover, comparison with similar work is also made in simulation to illustrate the effectiveness of the proposed strategy.