An apparatus comprises an input node, a power rail to power a circuit load, and multiple current paths coupled in parallel with each other between the input node and the power rail. Each current path respectively provides a sense output to indicate current in the path and a current switch having a control input to control the current in the path. A control circuit, coupled to each control input individually and to each sense output individually, controls the current in each path individually based on the indicated current therein after a non-zero input voltage is initially applied to the input node, such that all of the paths concurrently conduct current from the input node to the power rail and collectively cause a total inrush current and corresponding voltage at the power rail to gradually increase.