Integrated circuits are provided that have sensitive circuitry such as programmable polysilicon fuses. Electrostatic discharge (ESD) protection circuitry is provided that prevents damage or undesired programming of the sensitive circuitry in the presence of an electrostatic discharge event. The electrostatic discharge protection circuitry may have a power ESD device that limits the voltage level across the sensitive circuitry to a maximum voltage and that draws current away from the sensitive circuitry when exposed to ESD signals. The electrostatic discharge protection circuitry may also have an ESD margin circuit that helps to prevent current flow through the sensitive circuitry when the maximum voltage is applied across the sensitive circuitry.