Ultra-stable, tunable magnetic fields are desirable for a wide range of applications in medical imaging, electron microscopy, quantum science, and atomic physics. Superconducting magnets operated in persistent current mode, with device current flowing in a closed superconducting loop disconnected from a power source, are a common approach for applications with the most stringent requirements on temporal field stability. We present a method for active control of this persistent current by means of dynamic inductance change within the superconducting circuit. For a first realization of this general technique, we consider a variable superconducting inductor placed in series with the main magnet. The inductor acts as a dynamic flux storage device capable of transferring flux to or from the main magnet through inductance change. This allows for fine and fast adjustments of the persistent current without the use of thermal switches that limit the speed and accuracy of many present-day methods. With first experiments employing this technique, we demonstrate stabilization of a 1.95 T Nb-Ti round lens for electron microscopy against decay resulting from residual losses in the superconducting circuit, and more generally show flexibility for precise control over the magnitude and waveform of the persistent current.