Phoenix ash is remnants of a Phoenix. It is able to kill Eve as evidenced by Dean when he used it to kill Eve. As Bobby puts it, "ashes of a Phoenix can burn the Mother". It's the only substance that can kill her, except by certain beings and weapons.
Dean drinks a shot of alcohol containing the ashes, thus introducing the Phoenix ashes into his blood. He then goads Eve into biting him. Eve is poisoned by the ashes and dies.