This paper is concerned with the event-triggered consensus problem for leader-following multi-agent systems under asynchronous denial-of-service (DoS) attacks. Since the asynchronous attacks on different edges may enlarge the defense burden, the topology connectivity is employed to reveal the effectiveness of attacks and determine the unhealthy time periods. Relying on the internal system state, a dynamic event-triggered mechanism is built to regulate the information transmission between agents. In this mechanism, a variable threshold weight composed of the deviations of the relative neighboring errors and historical state errors is proposed to adaptively schedule the triggering thresholds with systems running stages. To guarantee secure consensus performance even during the triggering intervals, the local estimation of the neighboring information is integrated into the controller such that sufficient conditions to obtain the control parameters and the tolerable attack parameters are established without any Zeno behavior. The proposed method is validated by a simulation study.