This paper considers the global exponential synchronization problem of two memristive chaotic recurrent neural networks with time-varying delays using periodically alternate output feedback control. First, the periodically alternate output feedback control rule is designed for the global exponential synchronization of two memristive chaotic recurrent neural networks. Then, according to the Lyapunov stability theory, we construct an appropriate Lyapunov-Krasovskii functional to derive several new sufficient conditions guaranteeing exponential synchronization of two memristive chaotic recurrent neural networks under periodically alternate output feedback control. Compared with existing results on synchronization conditions on the basis of linear matrix inequalities of memristive chaotic recurrent neural networks, the derived results complement, extend earlier related results, and are also easy to validate in this paper. An illustrative example is provided to illustrate the effectiveness of the synchronization criteria.