Quantum systems with exact analytic solutions are rare, with most systems of interest requiring perturbative or other approximate methods to produce theoretical models. Husimi's 1953 solution to the Schrödinger equation for the linearly driven (see-saw) harmonic oscillator results in an exact solution for a wavepacket spatially translated but otherwise unperturbed by the driving force. In this work, we consider this Husimi driving scheme applied to a Bose-Einstein condensate (BEC), and further extend the theoretical solution to include interacting many-body systems in arbitrary states. We experimentally implement this solution in both optically-and magnetically-trapped BEC systems, subject to resonant or off-resonant driving by a linear magnetic potential. The observed trajectories of the centre-of-mass agree with theory, showing minimal excitation of the condensate. Based on these results, we propose Husimi driving as a platform for precision control of one-body, few-body, and many-body systems, and demonstrate its application to high-fidelity condensate transport at 72 times faster than adiabatic speeds. We demonstrate the platform's utility in trap frequency measurement and introduce a Husimi driving-based scheme for atom interferometry.