Density-dependent Markov chains form an important class of continuous-time Markov chains in population dynamics. On any fixed time window [0, T ], when the scale parameter K > 0 is large such chains are well approximated by the solution of an ODE (the fluid limit), with Gaussian fluctuations superimposed upon it. In this paper we quantify the period of time during which this Gaussian approximation remains precise, uniformly on the trajectory, in the case where the fluid limit converges to an exponentially stable equilibrium point. We provide a new coupling between the density-dependent chain and the approximating Gaussian process, based on a construction of Kurtz using the celebrated Komlós-Major-Tusnády theorem for random walks. We show that under mild hypotheses the time T(K) necessary for the strong approximation error to reach a threshold ε(K)<<1 is at least of order exp(V K ε(K)), for some constant V > 0. This notably entails that the Gaussian approximation yields the correct asymptotics regarding the time scales of moderate deviations. We also present applications to the Gaussian approximation of the logistic birth-and-death process conditioned to survive, and to the estimation of a quantity modeling the cost of an epidemic.