In this paper, an exponential-attenuation-based switching event-trigger (EABSET) scheme is designed to achieve the global stabilization of delayed memristive neural networks (MNNs). The issue is proposed for two reasons: (1) the available methods may be complicated in dealing with the state-dependent memristive connection weights; (2) the existing event-trigger mechanisms may be conservative in decreasing the amount of triggering times. To overcome these difficulties, the stabilization problem is formulated within a framework of networked control first. Then, an exponential attenuation term is introduced into the prescribed threshold function. It can enlarge the time span between two neighboring triggered events and further reduce the frequency of data packets sending out. By utilizing the input delay approach, time-dependent and piecewise Lyapunov functionals, and matrix norm inequalities, some sufficient criteria are obtained to guarantee the global stabilization of delayed MNNs and to design both the controller and the trigger parameters. Finally, some comparison simulation results demonstrate that the novel event-trigger scheme has some advantages over some existing ones.