Techniques are described for avoiding traffic black-holing in a multi-homed Ethernet virtual private networks (EVPNs) in which a customer device (CE) is multi-homed to a plurality of multi-homing provider edge devices (PEs) via respective links of an Ethernet segment. An overlay network is created over the Ethernet segment, and the multi-homing PEs of the EVPN are configured with a common anycast IP address for respective virtual network interfaces. Upon election as active designated forwarder (DF) for the EVPN, the DF PE of the multi-homing PEs advertises toward the customer network an IGP metric for the anycast IP address that is lower than the IGP metric(s) advertised by any of the non-DF standby PE routers segment to direct the CE to forward network packets from the customer network to the DF PE over the respective link of the Ethernet segment.