Nonlinear differential systems with nonlinearities satisfying sector constraints and with constant delays are studied. Such systems belong to well-known class of Persidskii-type systems, and they are widely used for modeling automatic control systems and neural networks. With the aid of the Lyapunov direct method and original constructions of Lyapunov–Krasovskii functionals, we derive conditions of the stability preservation under discretization of the considered differential systems. The fulfilment of these conditions guarantees that the zero solutions of the corresponding difference systems are asymptotically stable for arbitrary values of delays. Moreover, estimates of the convergence rate of solutions are obtained. The proposed approaches are used for the stability analysis of a discrete-time model of population dynamics. An example is given to demonstrate the effectiveness of our results.