In this paper, we investigate the exponential stability for a class of impulsive stochastic functional differential equations with delayed impulse effects. The impulse instants considered in this paper are random times which are different from the existing literature. By using stochastic process theory, stochastic analysis theory, the Razumikhin technology and Lyapunov method, we derive some new criteria of the moment exponential stability for the suggested system. As an application, we obtain some new pinning impulsive control methods for the stabilisation of stochastic neural networks. Finally, some numerical examples are provided to illustrate the effectiveness of the obtained results.