Memristors are widely considered as promising elements for the efficient implementation of synaptic weights in artificial neural networks (ANNs) since they are resistors that keep memory of their previous conductive state. Whereas demonstrations of simple neural networks (e.g., a single-layer perceptron) based on memristors already exist, the implementation of more complicated networks is more challenging and has yet to be reported. In this study, we demonstrate linearly nonseparable combinational logic classification (XOR logic task) using a network implemented with CMOS-based neurons and organic memrisitive devices that constitutes the first step toward the realization of a double layer perceptron. We also show numerically the ability of such network to solve a principally analogue task which cannot be realized by digital devices. The obtained results prove the possibility to create a multilayer ANN based on memristive devices that paves the way for designing a more complex network such as the double layer perceptron.