For processing streaming events from a Dynamic Vision Sensor camera, two types of neural networks can be considered. One are spiking neural networks, where simple spike-based computation is suitable for low-power consumption, but the discontinuity in spikes can make the training complicated in terms of hardware. The other one are digital Complementary Metal Oxide Semiconductor (CMOS)-based neural networks that can be trained directly using the normal backpropagation algorithm. However, the hardware and energy overhead can be significantly large, because all streaming events must be accumulated and converted into histogram data, which requires a large amount of memory such as SRAM. In this paper, to combine the spike-based operation with the normal backpropagation algorithm, memristor–CMOS hybrid circuits are proposed for implementing event-driven neural networks in hardware. The proposed hybrid circuits are composed of input neurons, synaptic crossbars, hidden/output neurons, and a neural network’s controller. Firstly, the input neurons perform preprocessing for the DVS camera’s events. The events are converted to histogram data using very simple memristor-based latches in the input neurons. After preprocessing the events, the converted histogram data are delivered to an ANN implemented using synaptic memristor crossbars. The memristor crossbars can perform low-power Multiply–Accumulate (MAC) calculations according to the memristor’s current–voltage relationship. The hidden and output neurons can convert the crossbar’s column currents to the output voltages according to the Rectified Linear Unit (ReLU) activation function. The neural network’s controller adjusts the MAC calculation frequency according to the workload of the event computation. Moreover, the controller can disable the MAC calculation clock automatically to minimize unnecessary power consumption. The proposed hybrid circuits have been verified by circuit simulation for several event-based datasets such as POKER-DVS and MNIST-DVS. The circuit simulation results indicate that the neural network’s performance proposed in this paper is degraded by as low as 0.5% while saving as much as 79% in power consumption for POKER-DVS. The recognition rate of the proposed scheme is lower by 0.75% compared to the conventional one, for the MNIST-DVS dataset. In spite of this little loss, the power consumption can be reduced by as much as 75% for the proposed scheme.