Vanadium dioxide (VO2), an archetypal correlated-electron material, undergoes an insulator-metal transition near room temperature that exhibits electron-correlation-driven and structurally driven physics. Using ultrafast temperature- and fluence-dependent optical spectroscopy and x-ray scattering, we show that multiple interrelated electronic and structural processes in the nonequilibrium dynamics in VO2 can be disentangled in the time domain. Specifically, following intense subpicosecond terahertz (THz) electric-field excitation, a partial collapse of the insulating gap occurs within the first picosecond. At temperatures sufficiently close to the transition temperature and for THz peak fields above a threshold of approximately 1 MV/cm, this electronic reconfiguration initiates a change in lattice symmetry taking place on a slower timescale. We identify the kinetic energy increase of electrons tunneling in the strong electric field as the driving force, illustrating a promising method to control electronic and structural interactions in correlated materials on an ultrafast timescale.