For the consensus problem of time-varying multi-agent systems under time-varying topology connection, an event-triggered impulsive control protocol was proposed. In this protocol, for each agent, the controller would be updated only when the related state error exceeded a threshold, and the control inputs would be carried out only at the event triggering instants, and continuous communication between agents was avoided. This protocol would greatly reduce the cost of communication and control for network consensus. The sufficient conditions for the multi-agent systems with time-varying characteristics to achieve consensus under event-triggered impulsive control were analyzed based on the algebraic graph theory, Lyapunov stability and impulsive differential equation. At the same time, it was proved theoretically that there was no Zeno behavior in the event-triggered time sequences. Finally, the effectiveness of the obtained theoretical conclusion was verified through several numerical simulations.