In term of the event-trigger impulsive mechanism, this paper designs a new event triggering function based on the Lyapunov function, and the leader-following consensus of multi-agent systems with randomly occurring uncertainties and randomly occurring nonlinearities is studied. Different from the control method of artificially setting the impulse time sequence, the generation of the impulsive moment depends on the designed triggering function, and when the trigger condition is met, the impulsive control is activated to reduce unnecessary control times and resource consumption. Based on impulsive differential equation theory, algebraic graph theory, and Lyapunov stability theory, the sufficiency conditions that controlled multi-agent systems can achieve the leader-following consensus are given. Meanwhile, the Zeno behavior can be excluded. Finally, the feasibility of the obtained results is verified by a numerical example.