This paper investigates the exponential consensus problem for a class of nonlinear leader-following multi-agent systems using impulsive control, where impulses are generated by the event-triggered mechanism and are subjected to actuation delays. It is proved that Zeno behavior can be avoided, and by employing the linear matrix inequality technique, some sufficient conditions for realizing exponential consensus of the considered system are derived. Actuation delay is an important factor affecting the consensus of the system, and our results show that increasing the actuation delay can enlarge the lower bound of the triggering interval, while it harms the consensus. To demonstrate the validity of the obtained results, a numerical example is provided.