Systems of differential equations with nonlinearities of a sector type and almost periodic perturbations are studied. With the aid of the Lyapunov direct method, conditions are found under which the considered systems admit globally asymptotically stable almost periodic solutions. Moreover, it is shown that the proposed approach permits us to derive new convergence conditions for some models of neural networks and generalized Lotka–Volterra models of population dynamics. An example is presented to demonstrate the effectiveness of the obtained results.